A001586 -id:A001586

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A185418

Square array, read by antidiagonals, used to recursively calculate the Springer numbers A001586.

+20
6

1, 1, 1, 3, 3, 1, 11, 11, 5, 1, 57, 57, 27, 7, 1, 361, 361, 175, 51, 9, 1, 2763, 2763, 1353, 413, 83, 11, 1, 24611, 24611, 12125, 3801, 819, 123, 13, 1, 250737, 250737, 123987, 39487, 8857, 1441, 171, 15, 1, 2873041, 2873041, 1424215, 458331, 105489, 18057, 2327, 227, 17, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The table entries T(n,k), n,k>=0, are defined by the recurrence relation:

1)... T(n+1,k) = k*T(n,k-1)+(k+1)*T(n,k+1) with boundary condition T(0,k) = 1.

The first column of the table produces the sequence of Springer numbers A001586.

For similarly defined tables see A185414, A185416 and A185420.

LINKS

Table of n, a(n) for n=0..54.

FORMULA

(1)... T(n,k) = S(n,k) with S(n,x) the polynomials described in A185417.

(2)... First column: T(n,0) = A001586(n).

(3)... Second column: T(n,1) = A001586(n+1).

(4)... Second row: T(1,k) = A005408(k).

(5)... Third row: T(2,k) = A164897(k).

EXAMPLE

Square array begins

n\k|.....0......1.......2.......3........4........5........6

============================================================

..0|.....1......1.......1.......1........1........1........1

..1|.....1......3.......5.......7........9.......11.......13

..2|.....3.....11......27......51.......83......123......171

..3|....11.....57.....175.....413......819.....1441.....2327

..4|....57....361....1353....3801.....8857....18057....33321

..5|...361...2763...12125...39487...105489...244211...507013

..6|..2763..24611..123987..458331..1379003..3569523..8229891

Examples of recurrence relation:

T(4,3) = 3801 = 3*T(3,2) + 4*T(3,4) = 3*175 + 4*819;

T(5,1) = 2763 = 1*T(4,0)+ 2*T(4,2) = 1*57 + 2*1353.

MAPLE

# A185418

S := proc(n, x) option remember; description `polynomials S(n, x)`;

if n = 0 then 1 else x*S(n-1, x-1)+(x+1)*S(n-1, x+1) end if end proc:

for n from 0 to 10 do seq(S(n, k), k = 0..10) end do;

MATHEMATICA

T[n_, k_] := T[n, k] = If[n<0 || k<0, 0, If[n == 0, 1, k T[n-1, k-1] + (k+1)*T[n-1, k+1]]];

Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 22 2021 *)

PROG

(PARI) {T(n, k)=if(n<0|k<0, 0, if(n==0, 1, k*T(n-1, k-1)+(k+1)*T(n-1, k+1)))}

CROSSREFS

Cf. A001586, A185417, A185414, A185416, A185420.

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Jan 30 2011

STATUS

approved

A214554

Triangle read by rows, coefficients of polynomials related to the Springer numbers A001586.

+20
0

1, -1, 2, -3, -4, 4, 11, -18, -12, 8, 57, 88, -72, -32, 16, -361, 570, 440, -240, -80, 32, -2763, -4332, 3420, 1760, -720, -192, 64, 24611, -38682, -30324, 15960, 6160, -2016, -448, 128, 250737, 393776, -309456, -161728, 63840, 19712, -5376, -1024, 256

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The polynomials might be called Springer polynomials because both p{n}(0) and p{n}(1) are signed versions of the Springer numbers. p{n}(0) is the first column of the triangle (A212435 with e.g.f. exp(-x)/cosh(2x)) and p{n}(1) are the row sums (A188458 with e.g.f. exp(x)/cosh(2x)).

LINKS

Table of n, a(n) for n=0..44.

FORMULA

p{0}(x) = 1 and for n>0

p{n}(x) = Sum_{0<=k<n; 2|k} 2^(n-k)*p{k}(1/2)*binomial(n,k)*((x-1/2)^(n-k)+(n mod 2)-1).

EXAMPLE

[0] 1,

[1] -1, 2,

[2] -3, -4, 4,

[3] 11, -18, -12, 8,

[4] 57, 88, -72, -32, 16,

[5] -361, 570, 440, -240, -80, 32,

[6] -2763, -4332, 3420, 1760, -720, -192, 64,

[7] 24611, -38682, -30324, 15960, 6160, -2016, -448, 128.

PROG

(Sage)

@CachedFunction

def SpringerPoly(n, x) :

if n == 0 : return 1

return add(2^(n-k)*SpringerPoly(k, 1/2)*binomial(n, k)*((x-1/2)^(n-k)+n%2-1) for k in range(n)[::2])

R = PolynomialRing(ZZ, 'x')

def A214554_row(n) : return R(SpringerPoly(n, x)).coeffs()

for n in (0..7) : A214554_row(n)

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Jul 30 2012

STATUS

approved

A256679

A signed triangle of V. I. Arnold for the Springer numbers (A001586).

+20
0

1, 1, 0, -2, -3, -3, -8, -6, -3, 0, 40, 48, 54, 57, 57, 256, 216, 168, 114, 57, 0, -1952, -2208, -2424, -2592, -2706, -2763, -2763, -17408, -15456, -13248, -10824, -8232, -5526, -2763, 0, 177280, 194688, 210144, 223392, 234216, 242448, 247974, 250737, 250737

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

This triangle is denoted R(b) on page 6 of the Arnold reference.

Unsigned version of triangle is triangle of A202818.

First column (m=0) is A000828. - Robert Israel, Apr 08 2015

LINKS

Table of n, a(n) for n=0..44.

V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups (in Russian), Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51. (See page 6)

FORMULA

E.g.f.: cosh(x+y)/cosh(2*(x+y))*exp(x).

T(n,m) = Sum_{k=floor(m/2)..floor(n/2)} Sum_{i=0..k}(4^i*E(2*i)*C(2*k,2*i))*C(n-m, 2*k-m)), where E(n) are the Euler secant numbers A122045.

EXAMPLE

1, 0;

-2, -3, -3;

-8, -6, -3, 0;

40, 48, 54, 57, 57;

256, 216, 168, 114, 57, 0;

MAPLE

T:= (n, m) -> add(add(4^i*euler(2*i)*binomial(2*k, 2*i)*binomial(n-m, 2*k-m), i=0..k), k=floor(m/2)..floor(n/2)):

seq(seq(T(n, m), m=0..n), n=0..10); # Robert Israel, Apr 08 2015

# Second program, about 100 times faster than the first for the first 100 rows.

Triangle := proc(len) local s, A, n, k;

A := Array(0..len-1, [1]); lprint(A[0]);

for n from 1 to len-1 do

if n mod 2 = 1 then s := 0 else

s := 2^(3*n+1)*(Zeta(0, -n, 1/8)-Zeta(0, -n, 5/8)) fi;

A[n] := s;

for k from n-1 by -1 to 0 do

s := s + A[k]; A[k] := s od;

lprint(seq(A[k], k=0..n));

od end:

Triangle(100); # Peter Luschny, Apr 08 2015

MATHEMATICA

T[n_, m_] := Sum[4^i EulerE[2i] Binomial[2k, 2i] Binomial[n-m, 2k-m], {k, Floor[m/2], n/2}, {i, 0, k}];

Table[T[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

PROG

(Maxima)

T(n, m):=(sum((sum(4^i*euler(2*i)*binomial(2*k, 2*i), i, 0, k))*binomial(n-m, 2*k-m), k, floor(m/2), n/2));

(Sage)

def triangle(len):

L = [1]; print(L)

for n in range(1, len):

if is_even(n):

s = 2^(3*n+1)*(hurwitz_zeta(-n, 1/8)-hurwitz_zeta(-n, 5/8))

else: s = 0

L.append(s)

for k in range(n-1, -1, -1):

s = s + L[k]; L[k] = s

print(L)

triangle(7) # Peter Luschny, Apr 08 2015

CROSSREFS

Cf. A000828, A001586, A002436, A008280, A122045, A202818, A256665.

KEYWORD

sign,tabl

AUTHOR

Vladimir Kruchinin, Apr 07 2015

STATUS

approved

A000111

Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250).
(Formerly M1492 N0587)

+10
335

1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, 370371188237525, 4951498053124096, 69348874393137901, 1015423886506852352, 15514534163557086905, 246921480190207983616, 4087072509293123892361

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Number of linear extensions of the "zig-zag" poset. See ch. 3, prob. 23 of Stanley. - Mitch Harris, Dec 27 2005

Number of increasing 0-1-2 trees on n vertices. - David Callan, Dec 22 2006

Also the number of refinements of partitions. - Heinz-Richard Halder (halder.bichl(AT)t-online.de), Mar 07 2008

The ratio a(n)/n! is also the probability that n numbers x1,x2,...,xn randomly chosen uniformly and independently in [0,1] satisfy x1 > x2 < x3 > x4 < ... xn. - Pietro Majer, Jul 13 2009

For n >= 2, a(n-2) = number of permutations w of an ordered n-set {x_1 < ... x_n} with the following properties: w(1) = x_n, w(n) = x_{n-1}, w(2) > w(n-1), and neither any subword of w, nor its reversal, has the first three properties. The count is unchanged if the third condition is replaced with w(2) < w(n-1). - Jeremy L. Martin, Mar 26 2010

A partition of zigzag permutations of order n+1 by the smallest or the largest, whichever is behind. This partition has the same recurrent relation as increasing 1-2 trees of order n, by induction the bijection follows. - Wenjin Woan, May 06 2011

As can be seen from the asymptotics given in the FORMULA section, one has lim_{n->oo} 2*n*a(n-1)/a(n) = Pi; see A132049/A132050 for the simplified fractions. - M. F. Hasler, Apr 03 2013

a(n+1) is the sum of row n in triangle A008280. - Reinhard Zumkeller, Nov 05 2013

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon (2011) give a far-reaching generalization of the bijection between Euler numbers and alternating permutations. - N. J. A. Sloane, Jul 09 2015

Number of treeshelves avoiding pattern T321. Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link, see A278678 for more definitions and examples. - Sergey Kirgizov, Dec 24 2016

Number of sequences (e(1), ..., e(n-1)), 0 <= e(i) < i, such that no three terms are equal. [Theorem 7 of Corteel, Martinez, Savage, and Weselcouch] - Eric M. Schmidt, Jul 17 2017

Number of self-dual edge-labeled trees with n vertices under "mind-body" duality. Also number of self-dual rooted edge-labeled trees with n vertices. See my paper linked below. - Nikos Apostolakis, Aug 01 2018

The ratio a(n)/n! is the volume of the convex polyhedron defined as the set of (x_1,...,x_n) in [0,1]^n such that x_i + x_{i+1} <= 1 for every 1 <= i <= n-1; see the solutions by Macdonald and Nelsen to the Amer. Math. Monthly problem referenced below. - Sanjay Ramassamy, Nov 02 2018

Number of total cyclic orders on {0,1,...,n} such that the triple (i-1,i,i+1) is positively oriented for every 1 <= i <= n-1; see my paper on cyclic orders linked below. - Sanjay Ramassamy, Nov 02 2018

The number of binary, rooted, unlabeled histories with n+1 leaves (following the definition of Rosenberg 2006). Also termed Tajima trees, Tajima genealogies, or binary, rooted, unlabeled ranked trees (Palacios et al. 2015). See Disanto & Wiehe (2013) for a proof. - Noah A Rosenberg, Mar 10 2019

From Gus Wiseman, Dec 31 2019: (Start)

Also the number of non-isomorphic balanced reduced multisystems with n + 1 distinct atoms and maximum depth. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The labeled version is A006472. For example, non-isomorphic representatives of the a(0) = 1 through a(4) = 5 multisystems are (commas elided):

{1} {12} {{1}{23}} {{{1}}{{2}{34}}} {{{{1}}}{{{2}}{{3}{45}}}}

{{{12}}{{3}{4}}} {{{{1}}}{{{23}}{{4}{5}}}}

{{{{1}{2}}}{{{3}}{{45}}}}

{{{{1}{23}}}{{{4}}{{5}}}}

{{{{12}}}{{{3}}{{4}{5}}}}

Also the number of balanced reduced multisystems with n + 1 equal atoms and maximum depth. This is possibly the meaning of Heinz-Richard Halder's comment (see also A002846, A213427, A265947). The non-maximum-depth version is A318813. For example, the a(0) = 1 through a(4) = 5 multisystems are (commas elided):

{1} {11} {{1}{11}} {{{1}}{{1}{11}}} {{{{1}}}{{{1}}{{1}{11}}}}

{{{11}}{{1}{1}}} {{{{1}}}{{{11}}{{1}{1}}}}

{{{{1}{1}}}{{{1}}{{11}}}}

{{{{1}{11}}}{{{1}}{{1}}}}

{{{{11}}}{{{1}}{{1}{1}}}}

(End)

With s_n denoting the sum of n independent uniformly random numbers chosen from [-1/2,1/2], the probability that the closest integer to s_n is even is exactly 1/2 + a(n)/(2*n!). (See Hambardzumyan et al. 2023, Appendix B.) - Suhail Sherif, Mar 31 2024

The number of permutations of size n+1 that require exactly n passes through a stack (i.e. have reverse-tier n-1) with an algorithm that prioritizes outputting the maximum possible prefix of the identity in a given pass and reverses the remainder of the permutation for prior to the next pass. - Rebecca Smith, Jun 05 2024

REFERENCES

M. D. Atkinson: Partial orders and comparison problems, Sixteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, (Boca Raton, Feb 1985), Congressus Numerantium 47, 77-88.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 34, 932.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 258-260, section #11.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262.

Bishal Deb and Alan D. Sokal, Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers, arXiv:2212.07232, Dec 14 2022.

H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 66.

O. Heimo and A. Karttunen, Series help-mates in 8, 9 and 10 moves (Problems 2901, 2974-2976), Suomen Tehtavaniekat (Proceedings of the Finnish Chess Problem Society) vol. 60, no. 2/2006, pp. 75, 77.

S. K. Jha, A Congruence for the Number of Alternating Permutations, Missouri J. Math. Sci., 33 (No. 1, 2021), 99-104

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C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 184.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997 and Vol. 2, 1999; see Problem 5.7.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..485 (terms 0..199 from N. J. A. Sloane)

Désiré André, Mémoire sur les permutations alternées, J. Math. Pur. Appl., 7 (1881), 167-184.

Nikos Apostolakis, A duality for labeled graphs and factorizations with applications to graph embeddings and Hurwitz enumeration, arXiv:1804.01214 [math.CO], 2018.

Joerg Arndt, Matters Computational (The Fxtbook), pp. 281-282.

V. I. Arnold, Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics, Duke Math. J. 63 (1991), 537-555.

J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.

M. D. Atkinson, Zigzag permutations and comparisons of adjacent elements, Information Processing Letters 21 (1985), 187-189.

Stefano Barbero, Umberto Cerruti, and Nadir Murru, Some combinatorial properties of the Hurwitz series ring arXiv:1710.05665 [math.NT], 2017.

Jean-Luc Baril, Sergey Kirgizov and Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.

Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.

B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.

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Graham Brightwell, Gérard Cohen, Emanuela Fachini, Marianne Fairthorne, János Körner, Gábor Simonyi and Ágnes Tóth, Permutation capacities of families of oriented infinite paths, SIAM J. Discrete Math. 24 (2010), no. 2, 441-456.

David Callan, A note on downup permutations and increasing 0-1-2 trees.

Peter J. Cameron and Liam Stott, Trees and cycles, arXiv:2010.14902 [math.CO], 2020. See p. 12.

Lorenzo Cappello and Julia A. Palacios, Sequential importance sampling for multi-resolution Kingman-Tajima coalescent counting, arXiv:1902.05527 [stat.AP], 2019.

Swee Hong Chan and Igor Pak, Computational complexity of counting coincidences, arXiv:2308.10214 [math.CO], 2023. See p. 16.

Xiao-Min Chen, X.-K. Chang, J.-Q. Sun, X./-B. Hu and Y.-N. Yeh, Three semi-discrete integrable systems related to orthogonal polynomials and their generalized determinant solutions, Nonlinearity, Volume 28, Number 7, Jun 08 2015.

Suyoung Choi, B. Park and H. Park, The Betti numbers of real toric varieties associated to Weyl chambers of type B, arXiv preprint arXiv:1602.05406 [math.AT], 2016.

Suyoung Choi and Younghan Yoon, The cohomology rings of real permutohedral varieties, arXiv:2308.12693 [math.AT], 2023.

Sean Cleary, Mareike Fischer, Robert C. Griffiths and Raazesh Sainudiin, Some distributions on finite rooted binary trees, UCDMS Research Report NO. UCDMS2015/2, School of Mathematics and Statistics, University of Canterbury, Christchurch, NZ, 2015.

C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.

Jane Ivy Coons and Seth Sullivant, The Cavender-Farris-Neyman Model with a Molecular Clock, arXiv:1805.04175 [math.AG], 2018.

Jane Ivy Coons and Seth Sullivant, The h*-polynomial of the order polytope of the zig-zag poset, arXiv:1901.07443 [math.CO], 2019.

Sylvie Corteel, Megan A. Martinez, Carla D. Savage and Michael Weselcouch, Patterns in Inversion Sequences I, arXiv:1510.05434 [math.CO], 2015

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Peter Luschny, Approximation, inclusion and asymptotics of the Euler numbers.

Peter Luschny, An old operation on sequences: the Seidel transform

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Index entries for "core" sequences

Index entries for sequences related to boustrophedon transform

FORMULA

E.g.f.: (1+sin(x))/cos(x) = tan(x) + sec(x).

E.g.f. for a(n+1) is 1/(cos(x/2) - sin(x/2))^2 = 1/(1-sin(x)) = d/dx(sec(x) + tan(x)).

E.g.f. A(x) = -log(1-sin(x)), for a(n+1). - Vladimir Kruchinin, Aug 09 2010

O.g.f.: A(x) = 1+x/(1-x-x^2/(1-2*x-3*x^2/(1-3*x-6*x^2/(1-4*x-10*x^2/(1-... -n*x-(n*(n+1)/2)*x^2/(1- ...)))))) (continued fraction). - Paul D. Hanna, Jan 17 2006

E.g.f. A(x) = y satisfies 2y' = 1 + y^2. - Michael Somos, Feb 03 2004

a(n) = P_n(0) + Q_n(0) (see A155100 and A104035), defining Q_{-1} = 0. Cf. A156142.

2*a(n+1) = Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k).

Asymptotics: a(n) ~ 2^(n+2)*n!/Pi^(n+1). For a proof, see for example Flajolet and Sedgewick.

a(n) = (n-1)*a(n-1) - Sum_{i=2..n-2} (i-1)*E(n-2, n-1-i), where E are the Entringer numbers A008281. - Jon Perry, Jun 09 2003

a(2*k) = (-1)^k euler(2k) and a(2k-1) = (-1)^(k-1)2^(2k)(2^(2k)-1) Bernoulli(2k)/(2k). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005

|a(n+1) - 2*a(n)| = A000708(n). - Philippe Deléham, Jan 13 2007

a(n) = 2^n|E(n,1/2) + E(n,1)| where E(n,x) are the Euler polynomials. - Peter Luschny, Jan 25 2009

a(n) = 2^(n+2)*n!*S(n+1)/(Pi)^(n+1), where S(n) = Sum_{k = -inf..inf} 1/(4k+1)^n (see the Elkies reference). - Emeric Deutsch, Aug 17 2009

a(n) = i^(n+1) Sum_{k=1..n+1} Sum_{j=0..k} binomial(k,j)(-1)^j (k-2j)^(n+1) (2i)^(-k) k^{-1}. - Ross Tang (ph.tchaa(AT)gmail.com), Jul 28 2010

a(n) = sum((if evenp(n+k) then (-1)^((n+k)/2)*sum(j!*Stirling2(n,j)*2^(1-j)*(-1)^(n+j-k)*binomial(j-1,k-1),j,k,n) else 0),k,1,n), n>0. - Vladimir Kruchinin, Aug 19 2010

If n==1(mod 4) is prime, then a(n)==1(mod n); if n==3(mod 4) is prime, then a(n)==-1(mod n). - Vladimir Shevelev, Aug 31 2010

For m>=0, a(2^m)==1(mod 2^m); If p is prime, then a(2*p)==1(mod 2*p). - Vladimir Shevelev, Sep 03 2010

From Peter Bala, Jan 26 2011: (Start)

a(n) = A(n,i)/(1+i)^(n-1), where i = sqrt(-1) and {A(n,x)}n>=1 = [1,1+x,1+4*x+x^2,1+11*x+11*x^2+x^3,...] denotes the sequence of Eulerian polynomials.

Equivalently, a(n) = i^(n+1)*Sum_{k=1..n} (-1)^k*k!*Stirling2(n,k) * ((1+i)/2)^(k-1) = i^(n+1)*Sum_{k = 1..n} (-1)^k*((1+i)/2)^(k-1)* Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*j^n.

This explicit formula for a(n) can be used to obtain congruence results. For example, for odd prime p, a(p) = (-1)^((p-1)/2) (mod p), as noted by Vladimir Shevelev above.

For the corresponding type B results see A001586. For the corresponding results for plane increasing 0-1-2 trees see A080635.

For generalized Eulerian, Stirling and Bernoulli numbers associated with the zigzag numbers see A145876, A147315 and A185424, respectively. For a recursive triangle to calculate a(n) see A185414.

(End)

a(n) = I^(n+1)*2*Li_{-n}(-I) for n > 0. Li_{s}(z) is the polylogarithm. - Peter Luschny, Jul 29 2011

a(n) = 2*Sum_{m=0..(n-2)/2} 4^m*(Sum_{i=m..(n-1)/2} (i-(n-1)/2)^(n-1)*binomial(n-2*m-1,i-m)*(-1)^(n-i-1)), n > 1, a(0)=1, a(1)=1. - Vladimir Kruchinin, Aug 09 2011

a(n) = D^(n-1)(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1-x^2)*d/dx. Cf. A006154. a(n) equals the alternating sum of the nonzero elements of row n-1 of A196776. This leads to a combinatorial interpretation for a(n); for example, a(4*n+2) gives the number of ordered set partitions of 4*n+1 into k odd-sized blocks, k = 1 (mod 4), minus the number of ordered set partitions of 4*n+1 into k odd-sized blocks, k = 3 (mod 4). Cf A002017. - Peter Bala, Dec 06 2011

From Sergei N. Gladkovskii, Nov 14 2011 - Dec 23 2013: (Start)

Continued fractions:

E.g.f.: tan(x) + sec(x) = 1 + x/U(0); U(k) = 4k+1-x/(2-x/(4k+3+x/(2+x/U(k+1)))).

E.g.f.: for a(n+1) is E(x) = 1/(1-sin(x)) = 1 + x/(1 - x + x^2/G(0)); G(k) = (2*k+2)*(2*k+3)-x^2+(2*k+2)*(2*k+3)*x^2/G(k+1).

E.g.f.: for a(n+1) is E(x) = 1/(1-sin(x)) = 1/(1 - x/(1 + x^2/G(0))) ; G(k) = 8*k+6-x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)).

E.g.f.: for a(n+1) is E(x) = 1/(1 - sin(x)) = 1/(1 - x*G(0)); G(k) = 1 - x^2/(2*(2*k+1)*(4*k+3) - 2*x^2*(2*k+1)*(4*k+3)/(x^2 - 4*(k+1)*(4*k+5)/G(k+1))).

E.g.f.: for a(n+1) is E(x) = 1/(1 - sin(x)) = 1/(1 - x*G(0)) where G(k)= 1 - x^2/( (2*k+1)*(2*k+3) - (2*k+1)*(2*k+3)^2/(2*k+3 - (2*k+2)/G(k+1))).

E.g.f.: tan(x) + sec(x) = 1 + 2*x/(U(0)-x) where U(k) = 4k+2 - x^2/U(k+1).

E.g.f.: tan(x) + sec(x) = 1 + 2*x/(2*U(0)-x) where U(k) = 4*k+1 - x^2/(16*k+12 - x^2/U(k+1)).

E.g.f.: tan(x) + sec(x) = 4/(2-x*G(0))-1 where G(k) = 1 - x^2/(x^2 - 4*(2*k+1)*(2*k+3)/G(k+1)).

G.f.: 1 + x/Q(0), m=+4, u=x/2, where Q(k) = 1 - 2*u*(2*k+1) - m*u^2*(k+1)*(2*k+1)/(1 - 2*u*(2*k+2) - m*u^2*(k+1)*(2*k+3)/Q(k+1)).

G.f.: conjecture: 1 + T(0)*x/(1-x), where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*(k+1))*(1-x*(k+2))/T(k+1)).

E.g.f.: 1+ 4*x/(T(0) - 2*x), where T(k) = 4*(2*k+1) - 4*x^2/T(k+1):

E.g.f.: T(0)-1, where T(k) = 2 + x/(4*k+1 - x/(2 - x/( 4*k+3 + x/T(k+1)))). (End)

E.g.f.: tan(x/2 + Pi/4). - Vaclav Kotesovec, Nov 08 2013

Asymptotic expansion: 4*(2*n/(Pi*e))^(n+1/2)*exp(1/2+1/(12*n) -1/(360*n^3) + 1/(1260*n^5) - ...). (See the Luschny link.) - Peter Luschny, Jul 14 2015

From Peter Bala, Sep 10 2015: (Start)

The e.g.f. A(x) = tan(x) + sec(x) satisfies A''(x) = A(x)*A'(x), hence the recurrence a(0) = 1, a(1) = 1, else a(n) = Sum_{i = 0..n-2} binomial(n-2,i)*a(i)*a(n-1-i).

Note, the same recurrence, but with the initial conditions a(0) = 0 and a(1) = 1, produces the sequence [0,1,0,1,0,4,0,34,0,496,...], an aerated version of A002105. (End)

a(n) = A186365(n)/n for n >= 1. - Anton Zakharov, Aug 23 2016

From Peter Luschny, Oct 27 2017: (Start)

a(n) = abs(2*4^n*(H(((-1)^n - 3)/8, -n) - H(((-1)^n - 7)/8, -n))) where H(z, r) are the generalized harmonic numbers.

a(n) = (-1)^binomial(n + 1, 2)*2^(2*n + 1)*(zeta(-n, 1 + (1/8)*(-7 + (-1)^n)) - zeta(-n, 1 + (1/8)*(-3 + (-1)^n))). (End)

a(n) = i*(i^n*Li_{-n}(-i) - (-i)^n*Li_{-n}(i)), where i is the imaginary unit and Li_{s}(z) is the polylogarithm. - Peter Luschny, Aug 28 2020

Sum_{n>=0} 1/a(n) = A340315. - Amiram Eldar, May 29 2021

a(n) = n!*Re([x^n](1 + I^(n^2 - n)*(2 - 2*I)/(exp(x) + I))). - Peter Luschny, Aug 09 2021

EXAMPLE

G.f. = 1 + x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 61*x^6 + 272*x^7 + 1385*x^8 + ...

Sequence starts 1,1,2,5,16,... since possibilities are {}, {A}, {AB}, {ACB, BCA}, {ACBD, ADBC, BCAD, BDAC, CDAB}, {ACBED, ADBEC, ADCEB, AEBDC, AECDB, BCAED, BDAEC, BDCEA, BEADC, BECDA, CDAEB, CDBEA, CEADB, CEBDA, DEACB, DEBCA}, etc. - Henry Bottomley, Jan 17 2001

MAPLE

A000111 := n-> n!*coeff(series(sec(x)+tan(x), x, n+1), x, n);

s := series(sec(x)+tan(x), x, 100): A000111 := n-> n!*coeff(s, x, n);

A000111:=n->piecewise(n mod 2=1, (-1)^((n-1)/2)*2^(n+1)*(2^(n+1)-1)*bernoulli(n+1)/(n+1), (-1)^(n/2)*euler(n)):seq(A000111(n), n=0..30); A000111:=proc(n) local k: k:=floor((n+1)/2): if n mod 2=1 then RETURN((-1)^(k-1)*2^(2*k)*(2^(2*k)-1)*bernoulli(2*k)/(2*k)) else RETURN((-1)^k*euler(2*k)) fi: end:seq(A000111(n), n=0..30); (C. Ronaldo)

T := n -> 2^n*abs(euler(n, 1/2)+euler(n, 1)): # Peter Luschny, Jan 25 2009

S := proc(n, k) option remember; if k=0 then RETURN(`if`(n=0, 1, 0)) fi; S(n, k-1)+S(n-1, n-k) end:

A000364 := n -> S(2*n, 2*n);

A000182 := n -> S(2*n+1, 2*n+1);

A000111 := n -> S(n, n); # Peter Luschny, Jul 29 2009

a := n -> 2^(n+2)*n!*(sum(1/(4*k+1)^(n+1), k = -infinity..infinity))/Pi^(n+1):

1, seq(a(n), n = 1..22); # Emeric Deutsch, Aug 17 2009

# alternative Maple program:

b:= proc(u, o) option remember;

`if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))

end:

a:= n-> b(n, 0):

seq(a(n), n=0..30); # Alois P. Heinz, Nov 29 2015

MATHEMATICA

n=22; CoefficientList[Series[(1+Sin[x])/Cos[x], {x, 0, n}], x] * Table[k!, {k, 0, n}] (* Jean-François Alcover, May 18 2011, after Michael Somos *)

a[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n+1)*(2^(n+1)-1)*BernoulliB[n+1])/(n+1)]]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 09 2012, after C. Ronaldo *)

ee = Table[ 2^n*EulerE[n, 1] + EulerE[n] - 1, {n, 0, 26}]; Table[ Differences[ee, n] // First // Abs, {n, 0, 26}] (* Jean-François Alcover, Mar 21 2013, after Paul Curtz *)

a[ n_] := If[ n < 0, 0, (2 I)^n If[ EvenQ[n], EulerE[n, 1/2], EulerE[n, 0] I]]; (* Michael Somos, Aug 15 2015 *)

a[ n_] := If[ n < 1, Boole[n == 0], With[{m = n - 1}, m! SeriesCoefficient[ 1 / (1 - Sin[x]), {x, 0, m}]]]; (* Michael Somos, Aug 15 2015 *)

s[0] = 1; s[_] = 0; t[n_, 0] := s[n]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0](* Jean-François Alcover, Feb 12 2016 *)

a[n_] := If[n == 0, 1, 2*Abs[PolyLog[-n, I]]]; (* Jean-François Alcover, Dec 02 2023, after M. F. Hasler *)

a[0] := 1; a[1] := 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] a[k] a[n - 1 - k], {k, 0, n - 2}]; Map[a, Range[0, 26]] (* Oliver Seipel, May 24 2024 after Peter Bala *)

a[0] := 1; a[1] := 1; a[n_] := a[n] = 1/(n (n-1)) Sum[a[n-1-k] a[k] k, {k, 1, n-1}]; Map[#! a[#]&, Range[0, 26]] (* Oliver Seipel, May 27 2024 *)

PROG

(PARI) {a(n) = if( n<1, n==0, n--; n! * polcoeff( 1 / (1 - sin(x + x * O(x^n))), n))}; \\ Michael Somos, Feb 03 2004

(PARI) {a(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+2, t=0; v = vector(k, i, if( i>1, t+= v[k+1-i]))); v[2])}; \\ Michael Somos, Feb 03 2004

(PARI) {a(n) = local(an); if( n<1, n>=0, an = vector(n+1, m, 1); for( m=2, n, an[m+1] = sum( k=0, m-1, binomial(m-1, k) * an[k+1] * an[m-k]) / 2); an[n+1])}; \\ Michael Somos, Feb 03 2004

(PARI) z='z+O('z^66); egf = (1+sin(z))/cos(z); Vec(serlaplace(egf)) \\ Joerg Arndt, Apr 30 2011

(PARI) A000111(n)={my(k); sum(m=0, n\2, (-1)^m*sum(j=0, k=n+1-2*m, binomial(k, j)*(-1)^j*(k-2*j)^(n+1))/k>>k)} \\ M. F. Hasler, May 19 2012

(PARI) A000111(n)=if(n, 2*abs(polylog(-n, I)), 1) \\ M. F. Hasler, May 20 2012

(Maxima) a(n):=sum((if evenp(n+k) then (-1)^((n+k)/2)*sum(j!*stirling2(n, j)*2^(1-j)*(-1)^(n+j-k)*binomial(j-1, k-1), j, k, n) else 0), k, 1, n); /* Vladimir Kruchinin, Aug 19 2010 */

(Maxima)

a(n):=if n<2 then 1 else 2*sum(4^m*(sum((i-(n-1)/2)^(n-1)*binomial(n-2*m-1, i-m)*(-1)^(n-i-1), i, m, (n-1)/2)), m, 0, (n-2)/2); /* Vladimir Kruchinin, Aug 09 2011 */

(Sage) # Algorithm of L. Seidel (1877)

def A000111_list(n) :

R = []; A = {-1:0, 0:1}; k = 0; e = 1

for i in (0..n) :

Am = 0; A[k + e] = 0; e = -e

for j in (0..i) : Am += A[k]; A[k] = Am; k += e

R.append(Am)

return R

A000111_list(22) # Peter Luschny, Mar 31 2012 (revised Apr 24 2016)

(Haskell)

a000111 0 = 1

a000111 n = sum $ a008280_row (n - 1)

-- Reinhard Zumkeller, Nov 01 2013

(Python)

# requires python 3.2 or higher

from itertools import accumulate

A000111_list, blist = [1, 1], [1]

for n in range(10**2):

blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))

A000111_list.append(sum(blist)) # Chai Wah Wu, Jan 29 2015

(Python)

from mpmath import *

mp.dps = 150

l = chop(taylor(lambda x: sec(x) + tan(x), 0, 26))

[int(fac(i) * li) for i, li in enumerate(l)] # Indranil Ghosh, Jul 06 2017

CROSSREFS

Cf. A000364 (secant numbers), A000182 (tangent numbers).

Related triangles: A008280, A008281, A008282, A008970, A010094, A059720, A177762.

Cf. A181937 for n-alternating permutations.

Cf. A109449 for an extension to an exponential Riordan array.

Column k=1 of A010094, A229892, A258829, A262124, A275784.

Column k=2 of A250261.

Cf. A002105, A186365.

For 0-1-2 trees with n nodes and k leaves, see A301344.

Matula-Goebel numbers of 0-1-2 trees are A292050.

An overview over generalized Euler numbers gives A349264.

Cf. A000258, A000311, A002846, A005121, A318813, A320270, A330474, A330665, A330679, A340315.

KEYWORD

nonn,core,eigen,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by M. F. Hasler, Apr 04 2013

Title corrected by Geoffrey Critzer, May 18 2013

STATUS

approved

A003881

Decimal expansion of Pi/4.

+10
92

7, 8, 5, 3, 9, 8, 1, 6, 3, 3, 9, 7, 4, 4, 8, 3, 0, 9, 6, 1, 5, 6, 6, 0, 8, 4, 5, 8, 1, 9, 8, 7, 5, 7, 2, 1, 0, 4, 9, 2, 9, 2, 3, 4, 9, 8, 4, 3, 7, 7, 6, 4, 5, 5, 2, 4, 3, 7, 3, 6, 1, 4, 8, 0, 7, 6, 9, 5, 4, 1, 0, 1, 5, 7, 1, 5, 5, 2, 2, 4, 9, 6, 5, 7, 0, 0, 8, 7, 0, 6, 3, 3, 5, 5, 2, 9, 2, 6, 6, 9, 9, 5, 5, 3, 7

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Also the ratio of the area of a circle to the circumscribed square. More generally, the ratio of the area of an ellipse to the circumscribed rectangle. Also the ratio of the volume of a cylinder to the circumscribed cube. - Omar E. Pol, Sep 25 2013

Also the surface area of a quarter-sphere of diameter 1. - Omar E. Pol, Oct 03 2013

Least positive solution to sin(x) = cos(x). - Franklin T. Adams-Watters, Jun 17 2014

Dirichlet L-series of the non-principal character modulo 4 (A101455) at 1. See e.g. Table 22 of arXiv:1008.2547. - R. J. Mathar, May 27 2016

This constant is also equal to the infinite sum of the arctangent functions with nested radicals consisting of square roots of two. Specifically, one of the Viete-like formulas for Pi is given by Pi/4 = Sum_{k = 2..oo} arctan(sqrt(2 - a_{k - 1})/a_k), where the nested radicals are defined by recurrence relations a_k = sqrt(2 + a_{k - 1}) and a_1 = sqrt(2) (see the article [Abrarov and Quine]). - Sanjar Abrarov, Jan 09 2017

Pi/4 is the area enclosed between circumcircle and incircle of a regular polygon of unit side. - Mohammed Yaseen, Nov 29 2023

REFERENCES

Jörg Arndt and Christoph Haenel, Pi: Algorithmen, Computer, Arithmetik, Springer 2000, p. 150.

Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, p. 408.

J. Rivaud, Analyse, Séries, équations différentielles, Mathématiques supérieures et spéciales, Premier cycle universitaire, Vuibert, 1981, Exercice 3, p. 136.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Sanjar M. Abrarov and Brendan M. Quine, A Viète-like formula for pi based on infinite sum of the arctangent functions with nested radicals, figshare, 4509014, (2017).

Peter Bala, Arctanh(z) and the Legendre polynomials

Jonathan M. Borwein, Peter B. Borwein, and Karl Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.

Ronald K. Hoeflin, Titan Test.

Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Literate Programs, Pi with Machin's formula (Haskell).

Michael Penn, A surprising appearance of pie!, YouTube video, 2020.

Michael Penn, Transforming normal identities into "crazy" ones, YouTube video, 2022.

Srinivasa Ramanujan, Question 353, J. Ind. Math. Soc.

Eric Weisstein's World of Mathematics, Prime Products.

Wikipedia, Leibniz formula for Pi.

Index entries for sequences from "Goedel, Escher, Bach".

Index entries for transcendental numbers.

FORMULA

Equals Integral_{x=0..oo} sin(2x)/(2x) dx.

Equals lim_{n->oo} n*A001586(n-1)/A001586(n) (conjecture). - Mats Granvik, Feb 23 2011

Equals Integral_{x=0..1} 1/(1+x^2) dx. - Gary W. Adamson, Jun 22 2003

Equals Integral_{x=0..Pi/2} sin(x)^2 dx, or Integral_{x=0..Pi/2} cos(x)^2 dx. - Jean-François Alcover, Mar 26 2013

Equals (Sum_{x=0..oo} sin(x)*cos(x)/x) - 1/2. - Bruno Berselli, May 13 2013

Equals (-digamma(1/4) + digamma(3/4))/4. - Jean-François Alcover, May 31 2013

Equals Sum_{n>=0} (-1)^n/(2*n+1). - Geoffrey Critzer, Nov 03 2013

Equals Integral_{x=0..1} Product_{k>=1} (1-x^(8*k))^3 dx [cf. A258414]. - Vaclav Kotesovec, May 30 2015

Equals Product_{k in A071904} (if k mod 4 = 1 then (k-1)/(k+1)) else (if k mod 4 = 3 then (k+1)/(k-1)). - Dimitris Valianatos, Oct 05 2016

From Peter Bala, Nov 15 2016: (Start)

For N even: 2*(Pi/4 - Sum_{k = 1..N/2} (-1)^(k-1)/(2*k - 1)) ~ (-1)^(N/2)*(1/N - 1/N^3 + 5/N^5 - 61/N^7 + 1385/N^9 - ...), where the sequence of unsigned coefficients [1, 1, 5, 61, 1385, ...] is A000364. See Borwein et al., Theorem 1 (a).

For N odd: 2*(Pi/4 - Sum_{k = 1..(N-1)/2} (-1)^(k-1)/(2*k - 1)) ~ (-1)^((N-1)/2)*(1/N - 1/N^2 + 2/N^4 - 16/N^6 + 272/N^8 - ...), where the sequence of unsigned coefficients [1, 1, 2, 16, 272, ...] is A000182 with an extra initial term of 1.

For N = 0,1,2,... and m = 1,3,5,... there holds Pi/4 = (2*N)! * m^(2*N) * Sum_{k >= 0} ( (-1)^(N+k) * 1/Product_{j = -N..N} (2*k + 1 + 2*m*j) ); when N = 0 we get the Madhava-Gregory-Leibniz series for Pi/4.

For examples of asymptotic expansions for the tails of these series representations for Pi/4 see A024235 (case N = 1, m = 1), A278080 (case N = 2, m = 1) and A278195 (case N = 3, m = 1).

For N = 0,1,2,..., Pi/4 = 4^(N-1)*N!/(2*N)! * Sum_{k >= 0} 2^(k+1)*(k + N)!* (k + 2*N)!/(2*k + 2*N + 1)!, follows by applying Euler's series transformation to the above series representation for Pi/4 in the case m = 1. (End)

From Peter Bala, Nov 05 2019: (Start)

For k = 0,1,2,..., Pi/4 = k!*Sum_{n = -oo..oo} 1/((4*n+1)*(4*n+3)* ...*(4*n+2*k+1)), where Sum_{n = -oo..oo} f(n) is understood as lim_{j -> oo} Sum_{n = -j..j} f(n).

Equals Integral_{x = 0..oo} sin(x)^4/x^2 dx = Sum_{n >= 1} sin(n)^4/n^2, by the Abel-Plana formula.

Equals Integral_{x = 0..oo} sin(x)^3/x dx = Sum_{n >= 1} sin(n)^3/n, by the Abel-Plana formula. (End)

From Amiram Eldar, Aug 19 2020: (Start)

Equals arcsin(1/sqrt(2)).

Equals Product_{k>=1} (1 - 1/(2*k+1)^2).

Equals Integral_{x=0..oo} x/(x^4 + 1) dx.

Equals Integral_{x=0..oo} 1/(x^2 + 4) dx. (End)

With offset 1, equals 5 * Pi / 2. - Sean A. Irvine, Aug 19 2021

Equals (1/2)!^2 = Gamma(3/2)^2. - Gary W. Adamson, Aug 23 2021

Equals Integral_{x = 0..oo} exp(-x)*sin(x)/x dx (see Rivaud reference). - Bernard Schott, Jan 28 2022

From Amiram Eldar, Nov 06 2023: (Start)

Equals beta(1), where beta is the Dirichlet beta function.

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p)^(-1). (End)

Equals arctan( F(1)/F(4) ) + arctan( F(2)/F(3) ), where F(1), F(2), F(3), and F(4) are any four consecutive Fibonacci numbers. - Gary W. Adamson, Mar 03 2024

Pi/4 = Sum_{n >= 1} i/(n*P(n, i)*P(n-1, i)) = (1/2)*Sum_{n >= 1} (-1)^(n+1)*4^n/(n*A006139(n)*A006139(n-1)), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The n-th summand of the series is O( 1/(3 + 2*sqrt(3))^n ). - Peter Bala, Mar 16 2024

Equals arctan( phi^(-3) ) + arctan(phi^(-1) ). - Gary W. Adamson, Mar 27 2024

EXAMPLE

0.785398163397448309615660845819875721049292349843776455243736148...

N = 2, m = 6: Pi/4 = 4!*3^4 Sum_{k >= 0} (-1)^k/((2*k - 11)*(2*k - 5)*(2*k + 1)*(2*k + 7)*(2*k + 13)). - Peter Bala, Nov 15 2016

MAPLE

evalf(Pi/4) ;

MATHEMATICA

RealDigits[N[Pi/4, 6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

(* PROGRAM STARTS *)

(* Define the nested radicals a_k by recurrence *)

a[k_] := Nest[Sqrt[2 + #1] & , 0, k]

(* Example of Pi/4 approximation at K = 100 *)

Print["The actual value of Pi/4 is"]

N[Pi/4, 40]

Print["At K = 100 the approximated value of Pi/4 is"]

K := 100; (* the truncating integer *)

N[Sum[ArcTan[Sqrt[2 - a[k - 1]]/a[k]], {k, 2, K}], 40] (* equation (8) *)

(* Error terms for Pi/4 approximations *)

Print["Error terms for Pi/4"]

k := 1; (* initial value of the index k *)

K := 10; (* initial value of the truncating integer K *)

sqn := {}; (* initiate the sequence *)

AppendTo[sqn, {"Truncating integer K ", " Error term in Pi/4"}];

While[K <= 30,

AppendTo[sqn, {K,

N[Pi/4 - Sum[ArcTan[Sqrt[2 - a[k - 1]]/a[k]], {k, 2, K}], 1000] //

N}]; K++]

Print[MatrixForm[sqn]]

(* Sanjar Abrarov, Jan 09 2017 *)

PROG

(Haskell) -- see link: Literate Programs

import Data.Char (digitToInt)

a003881_list len = map digitToInt $ show $ machin `div` (10 ^ 10) where

machin = 4 * arccot 5 unity - arccot 239 unity

unity = 10 ^ (len + 10)

arccot x unity = arccot' x unity 0 (unity `div` x) 1 1 where

arccot' x unity summa xpow n sign

| term == 0 = summa

| otherwise = arccot'

x unity (summa + sign * term) (xpow `div` x ^ 2) (n + 2) (- sign)

where term = xpow `div` n

-- Reinhard Zumkeller, Nov 20 2012

(SageMath) # Leibniz/Cohen/Villegas/Zagier/Arndt/Haenel

def FastLeibniz(n):

b = 2^(2*n-1); c = b; s = 0

for k in range(n-1, -1, -1):

t = 2*k+1

s = s + c/t if is_even(k) else s - c/t

b *= (t*(k+1))/(2*(n-k)*(n+k))

c += b

return s/c

A003881 = RealField(3333)(FastLeibniz(1330))

print(A003881) # Peter Luschny, Nov 20 2012

(PARI) Pi/4 \\ Charles R Greathouse IV, Jul 07 2014

(Magma) R:= RealField(100); Pi(R)/4; // G. C. Greubel, Mar 08 2018

CROSSREFS

Cf. A000796, A001586, A071904, A019669, A197723, A347152.

Cf. A000182, A000364, A024235, A278080, A278195.

Cf. A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A001622.

KEYWORD

nonn,cons,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

a(98) and a(99) corrected by Reinhard Zumkeller, Nov 20 2012

STATUS

approved

A153641

Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).

+10
53

1, 1, 1, -1, 1, -3, 1, -6, 5, 1, -10, 25, 1, -15, 75, -61, 1, -21, 175, -427, 1, -28, 350, -1708, 1385, 1, -36, 630, -5124, 12465, 1, -45, 1050, -12810, 62325, -50521, 1, -55, 1650, -28182, 228525, -555731, 1, -66, 2475, -56364, 685575, -3334386, 2702765, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

In the following the expression [n odd] is 1 if n is odd, 0 otherwise.

(+) W_n(0) = E_n are the Euler (or secant) numbers A122045.

(+) W_n(1) = T_n are the signed tangent numbers, see A009006.

(+) W_{n-1}(1) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli number A027641/A027642.

(+) W_n(-1) 2^{-n}(n+1) = G_n the Genocchi number A036968.

(+) W_n(1/2) 2^{n} are the signed generalized Euler (Springer) number, see A001586.

(+) | W_n([n odd]) | the number of alternating permutations A000111.

(+) | W_n([n odd]) / n! | for 0<=n the Euler zeta number A099612/A099617 (see Wikipedia on Bernoulli number). - Peter Luschny, Dec 29 2008

The diagonals in the full triangle (with zero coefficients) of the polynomials have the general form E(k)*binomial(n+k,k) (k>=0 fixed, n=0,1,...) where E(n) are the Euler numbers in the enumeration A122045. For k=2 we find the triangular numbers A000217 and for k=4 A154286. - Peter Luschny, Jan 06 2009

From Peter Bala, Jun 10 2009: (Start)

The Swiss-Knife polynomials W_n(x) may be expressed in terms of the Bernoulli polynomials B(n,x) as

... W_n(x) = 4^(n+1)/(2*n+2)*[B(n+1,(x+3)/4) - B(n+1,(x+1)/4)].

The Swiss-Knife polynomials are, apart from a multiplying factor, examples of generalized Bernoulli polynomials.

Let X be the Dirichlet character modulus 4 defined by X(4*n+1) = 1, X(4*n+3) = -1 and X(2*n) = 0. The generalized Bernoulli polynomials B(X;n,x), n = 1,2,..., associated with the character X are defined by means of the generating function

... t*exp(x*t)*(exp(t)-exp(3*t))/(exp(4*t)-1) = sum {n = 1..inf} B(X;n,x)*t^n/n!.

The first few values are B(X;1,x) = -1/2, B(X;2,x) = -x, B(X,3,x) = -3/2*(x^2-1) and B(X;4,x) = -2*(x^3-3*x).

In general, W_n(x) = -2/(n+1)*B(X;n+1,x).

For the theory of generalized Bernoulli polynomials associated to a periodic arithmetical function see [Cohen, Section 9.4].

The generalized Bernoulli polynomials may be used to evaluate twisted sums of k-th powers. For the present case the result is

sum{n = 0..4*N-1} X(n)*n^k = 1^k - 3^k + 5^k - 7^k + ... - (4*N-1)^k

= [B(X;k+1,4*N) - B(X;k+1,0)]/(k+1) = [W_k(0) - W_k(4*N)]/2.

For the proof apply [Cohen, Corollary 9.4.17 with m = 4 and x = 0].

The generalized Bernoulli polynomials and the Swiss-Knife polynomials are also related to infinite sums of powers through their Fourier series - see the formula section below. For a table of the coefficients of generalized Bernoulli polynomials attached to a Dirichlet character modulus 8 see A151751.

(End)

The Swiss-Knife polynomials provide a general formula for alternating sums of powers similar to the formula which are provided by the Bernoulli polynomials for non-alternating sums of powers (see the Luschny link). Sequences covered by this formula include A001057, A062393, A062392, A011934, A144129, A077221, A137501, A046092. - Peter Luschny, Jul 12 2009

The greatest common divisor of the nonzero coefficients of the decapitated Swiss-Knife polynomials is exp(Lambda(n)), where Lambda(n) is the von Mangoldt function for odd primes, symbolically:

gcd(coeffs(SKP_{n}(x) - x^n)) = A155457(n) (n>1). - Peter Luschny, Dec 16 2009

Another version is at A119879. - Philippe Deléham, Oct 26 2013

REFERENCES

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag. [From Peter Bala, Jun 10 2009]

LINKS

G. C. Greubel, Table of n, a(n) for the first 76 rows, flattened

Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.

Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.

Leonhard Euler (1735), De summis serierum reciprocarum, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415v2 (math.HO), 2005-2008.

A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414. [Added by Tom Copeland, Aug 31 2015]

Peter Luschny, The Swiss-Knife polynomials.

Peter Luschny, Swiss-Knife polynomials and Euler numbers

Wikipedia, Bernoulli number

J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

FORMULA

W_n(x) = Sum_{k=0..n}{v=0..k} (-1)^v binomial(k,v)*c_k*(x+v+1)^n where c_k = frac((-1)^(floor(k/4))/2^(floor(k/2))) [4 not div k] (Iverson notation).

From Peter Bala, Jun 10 2009: (Start)

E.g.f.: 2*exp(x*t)*(exp(t)-exp(3*t))/(1-exp(4*t))= 1 + x*t + (x^2-1)*t^2/2! + (x^3-3*x)*t^3/3! + ....

W_n(x) = 1/(2*n+2)*Sum_{k=0..n+1} 1/(k+1)*Sum_{i=0..k} (-1)^i*binomial(k,i)*((x+4*i+3)^(n+1) - (x+4*i+1)^(n+1)).

Fourier series expansion for the generalized Bernoulli polynomials:

B(X;2*n,x) = (-1)^n*(2/Pi)^(2*n)*(2*n)! * {sin(Pi*x/2)/1^(2*n) - sin(3*Pi*x/2)/3^(2*n) + sin(5*Pi*x/2)/5^(2*n) - ...}, valid for 0 <= x <= 1 when n >= 1.

B(X;2*n+1,x) = (-1)^(n+1)*(2/Pi)^(2*n+1)*(2*n+1)! * {cos(Pi*x/2)/1^(2*n+1) - cos(3*Pi*x/2)/3^(2*n+1) + cos(5*Pi*x/2)/5^(2*n+1) - ...}, valid for 0 <= x <= 1 when n >= 1 and for 0 <= x < 1 when n = 0.

(End)

E.g.f.: exp(x*t) * sech(t). - Peter Luschny, Jul 07 2009

O.g.f. as a J-fraction: z/(1-x*z+z^2/(1-x*z+4*z^2/(1-x*z+9*z^2/(1-x*z+...)))) = z + x*z^2 + (x^2-1)*z^3 + (x^3-3*x)*z^4 + .... - Peter Bala, Mar 11 2012

Conjectural o.g.f.: Sum_{n >= 0} 1/2^(n-1)/2)*cos((n+1)*Pi/4)*( Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - (k + x)*t) ) = 1 + x*t + (x^2 - 1)*t^2 + (x^3 - 3*x)*t^3 + ... (checked up to O(t^13)), which leads to W_n(x) = Sum_{k = 0..n} 1/2^((k - 1)/2)*cos((k + 1)*Pi/4)*( Sum_{j = 0..k} (-1)^j*binomial(k, j)*(j + x)^n ). - Peter Bala, Oct 03 2016

EXAMPLE

x^2 -1

x^3 -3x

x^4 -6x^2 +5

x^5 -10x^3 +25x

x^6 -15x^4 +75x^2 -61

x^7 -21x^5 +175x^3 -427x

MAPLE

w := proc(n, x) local v, k, pow, chen; pow := (a, b) -> if a = 0 and b = 0 then 1 else a^b fi; chen := proc(m) if irem(m+1, 4) = 0 then RETURN(0) fi; 1/((-1)^iquo(m+1, 4) *2^iquo(m, 2)) end; add(add((-1)^v*binomial(k, v)*pow(v+x+1, n)*chen(k), v=0..k), k=0..n) end:

# Coefficients with zeros:

seq(print(seq(coeff(i!*coeff(series(exp(x*t)*sech(t), t, 16), t, i), x, i-n), n=0..i)), i=0..8);

# Recursion

W := proc(n, z) option remember; local k, p;

if n = 0 then 1 else p := irem(n+1, 2);

z^n - p + add(`if`(irem(k, 2)=1, 0,

W(k, 0)*binomial(n, k)*(power(z, n-k)-p)), k=2..n-1) fi end:

# Peter Luschny, edited and additions Jul 07 2009, May 13 2010, Oct 24 2011

MATHEMATICA

max = 9; rows = (Reverse[ CoefficientList[ #, x]] & ) /@ CoefficientList[ Series[ Exp[x*t]*Sech[t], {t, 0, max}], t]*Range[0, max]!; par[coefs_] := (p = Partition[ coefs, 2][[All, 1]]; If[ EvenQ[ Length[ coefs]], p, Append[ p, Last[ coefs]]]); Flatten[ par /@ rows] (* Jean-François Alcover, Oct 03 2011, after g.f. *)

sk[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; Table[CoefficientList[sk[n, x], x] // Reverse // Select[#, # =!= 0 &] &, {n, 0, 13}] // Flatten (* Jean-François Alcover, May 21 2013 *)

PROG

(Sage)

def A046978(k):

if k % 4 == 0:

return 0

return (-1)**(k // 4)

def A153641_poly(n, x):

return expand(add(2**(-(k // 2))*A046978(k+1)*add((-1)**v*binomial(k, v)*(v+x+1)**n for v in (0..k)) for k in (0..n)))

for n in (0..7): print(A153641_poly(n, x)) # Peter Luschny, Oct 24 2011

CROSSREFS

Cf. A151751, A162590, A000111, A001586, A009006, A027641/A027642, A036968, A099612/A099617, A119879, A104033.

W_n(k), k=0,1,...

W_0: 1, 1, 1, 1, 1, 1, ........ A000012

W_1: 0, 1, 2, 3, 4, 5, ........ A001477

W_2: -1, 0, 3, 8, 15, 24, ........ A067998

W_3: 0, -2, 2, 18, 52, 110, ........ A121670

W_4: 5, 0, -3, 32, 165, 480, ........

W_n(k), n=0,1,...

k=0: 1, 0, -1, 0, 5, 0, -61, ... A122045

k=1: 1, 1, 0, -2, 0, 16, 0, ... A155585

k=2: 1, 2, 3, 2, -3, 2, 63, ... A119880

k=3: 1, 3, 8, 18, 32, 48, 128, ... A119881

k=4: 1, 4, 15, 52, 165, 484, ........ [Peter Luschny, Jul 07 2009]

KEYWORD

easy,sign,tabf

AUTHOR

Peter Luschny, Dec 29 2008

STATUS

approved

A046802

T(n, k) = Sum_{j=k..n} binomial(n, j)*E1(j, j-k), where E1 are the Eulerian numbers A173018. Triangle read by rows, T(n, k) for 0 <= k <= n.

+10
24

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 33, 15, 1, 1, 31, 131, 131, 31, 1, 1, 63, 473, 883, 473, 63, 1, 1, 127, 1611, 5111, 5111, 1611, 127, 1, 1, 255, 5281, 26799, 44929, 26799, 5281, 255, 1, 1, 511, 16867, 131275, 344551, 344551, 131275, 16867, 511, 1, 1, 1023, 52905

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

T(n,k) is the number of positroid cells of the totally nonnegative Grassmannian G+(k,n) (cf. Postnikov/Williams). It is the triangle of the h-vectors of the stellahedra. - Tom Copeland, Oct 10 2014

See A248727 for a simple transformation of the row polynomials of this entry that produces the umbral compositional inverses of the polynomials of A074909, related to the face polynomials of the simplices. - Tom Copeland, Jan 21 2015

From Tom Copeland, Jan 24 2015: (Start)

The reciprocal of this entry's e.g.f. is [t e^(-xt) - e^(-x)] / (t-1) = 1 - (1+t) x + (1+t+t^2) x^2/2! - (1+t+t^2+t^3) x^3/3! + ... = e^(q.(0;t)x), giving the base sequence (q.(0;t))^n = q_n(0;t) = (-1)^n [1-t^(n+1)] / (1-t) for the umbral compositional inverses (q.(0;t)+z)^n = q_n(z;t) of the Appell polynomials associated with this entry, p_n(z;t) below, i.e., q_n(p.(z;t)) = z^n = p_n(q.(z;t)), in umbral notation. The relations in A133314 then apply between the two sets of base polynomials. (Inserted missing index in a formula - Mar 03 2016.)

The associated o.g.f. for the umbral inverses is Og(x) = x / (1-x q.(0:t)) = x / [(1+x)(1+tx)] = x / [1+(1+t)x+tx^2]. Applying A134264 to h(x) = x / Og(x) = 1 + (1+t) x + t x^2 leads to an o.g.f. for the Narayana polynomials A001263 as the comp. inverse Oginv(x) = [1-(1+t)x-sqrt[1-2(1+t)x+((t-1)x)^2]] / (2xt). Note that Og(x) gives the signed h-polynomials of the simplices and that Oginv(x) gives the h-polynomials of the simplicial duals of the Stasheff polynomials, or type A associahedra. Contrast this with A248727 = A046802 * A007318, which has o.g.f.s related to the corresponding f-polynomials. (End)

The Appell polynomials p_n(x;t) in the formulas below specialize to the Swiss-knife polynomials of A119879 for t = -1, so the Springer numbers A001586 are given by 2^n p_n(1/2;-1). - Tom Copeland, Oct 14 2015

The row polynomials are the h-polynomials associated to the stellahedra, whose f-polynomials are the row polynomials of A248727. Cf. page 60 of the Buchstaber and Panov link. - Tom Copeland, Nov 08 2016

The row polynomials are the h-polynomials of the stellohedra, which enumerate partial permutations according to descents. Cf. Section 10.4 of the Postnikov-Reiner-Williams reference. - Lauren Williams, Jul 05 2022

From p. 60 of the Buchstaber and Panov link, S = P * C / T where S, P, C, and T are the bivariate e.g.f.s of the h vectors of the stellahedra, permutahedra, hypercubes, and (n-1)-simplices, respectively. - Tom Copeland, Jan 09 2017

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, Holland, 1974, page 245 [From Roger L. Bagula, Nov 21 2009]

D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)

Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.

Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

Paul Barry, Series reversion with Jacobi and Thron continued fractions, arXiv:2107.14278 [math.NT], 2021.

Carolina Benedetti, Anastasia Chavez, and Daniel Tamayo, Quotients of uniform positroids, arXiv:1912.06873 [math.CO], 2019.

V. Buchstaber and T. Panov Toric Topology, arXiv:1210.2368v3 [math.AT], 2014.

Jan Geuenich, Tilting modules for the Auslander algebra of K[x]/(xn), arXiv:1803.10707 [math.RT], 2018.

Jun Ma, Shi-mei Ma, Yeong-Nan Yeh, Recurrence relations for binomial-eulerian polynomials, arXiv:1711.09016, Thm. 2.1.

A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

A. Postnikov, V. Reiner, and L. Williams, Faces of generalized permutohedra, arXiv:math/0609184 [math.CO], 2006-2007.

D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70. [Annotated scanned copy]

L. K. Williams, Enumeration of totally positive Grassmann cells, arXiv:math/0307271 [math.CO], 2003-2004.

L. Williams, The Positive Grassmannian (from a mathematician's perspective), 2014

FORMULA

E.g.f.: (y-1)*exp(x*y)/(y-exp((y-1)*x)). - Vladeta Jovovic, Sep 20 2003

p(t,x) = (1 - x)*exp(t)/(1 - x*exp(t*(1 - x))). - Roger L. Bagula, Nov 21 2009

With offset=0, T(n,0)=1 otherwise T(n,k) = sum_{i=0..k-1} C(n,i)((i-k)^i*(k-i+1)^(n-i) - (i-k+1)^i*(k-i)^(n-i)) (cf. Williams). - Tom Copeland, Oct 10 2014

With offset 0, T = A007318 * A123125. Second column is A000225; 3rd, appears to be A066810. - Tom Copeland, Jan 23 2015

A raising operator (with D = d/dx) associated with this entry's row polynomials is R = x + t + (1-t) / [1-t e^{(1-t)D}] = x + t + 1 + t D + (t+t^2) D^2/2! + (t+4t^2+t^3) D^3/3! + ... , containing the e.g.f. for the Eulerian polynomials of A123125. Then R^n 1 = (p.(0;t)+x)^n = p_n(x;t) are the Appell polynomials with this entry's row polynomials p_n(0;t) as the base sequence. Examples of this formalism are given in A028246 and A248727. - Tom Copeland, Jan 24 2015

With offset 0, T = A007318*(padded A090582)*(inverse of A097805) = A007318*(padded A090582)*(padded A130595) = A007318*A123125 = A007318*(padded A090582)*A007318*A097808*A130595, where padded matrices are of the form of padded A007318, which is A097805. Inverses of padded matrices are just the padded versions of inverses of the unpadded matrices. This relates the face vectors, or f-vectors, and h-vectors of the permutahedra / permutohedra to those of the stellahedra / stellohedra. - Tom Copeland, Nov 13 2016

Umbrally, the row polynomials (offset 0) are r_n(x) = (1 + q.(x))^n, where (q.(x))^k = q_k(x) are the row polynomials of A123125. - Tom Copeland, Nov 16 2016

From the previous umbral statement, OP(x,d/dy) y^n = (y + q.(x))^n, where OP(x,y) = exp[y * q.(x)] = (1-x)/(1-x*exp((1-x)y)), the e.g.f. of A123125, so OP(x,d/dy) y^n evaluated at y = 1 is r_n(x), the n-th row polynomial of this entry, with offset 0. - Tom Copeland, Jun 25 2018

Consolidating some formulas in this entry and A248727, in umbral notation for concision, with all offsets 0: Let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of this entry (A046802, the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of A248727 (the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - Tom Copeland, Jan 24 2020

From Peter Luschny, Apr 30 2021: (Start)

Sum_{k=0..n} (-1)^k*T(n, k) = A122045(n).

Sum_{k=0..n} 2^(n-k)*T(n,k) = A007047(n).

Sum_{k=0..n} T(n, n-k) = A000522(n).

Sum_{k=0..n} T(n-k, k) = Sum_{k=0..n} (n - k)^k = A026898(n-1) for n >= 1.

Sum_{k=0..n} k*T(n, k) = A036919(n) = floor(n*n!*e/2).

(End)

EXAMPLE

The triangle T(n, k) begins:

n\k 0 1 2 3 4 5 6 7

0: 1

1: 1 1

2: 1 3 1

3: 1 7 7 1

4: 1 15 33 15 1

5: 1 31 131 131 31 1

6: 1 63 473 883 473 63 1

7: 1 127 1611 5111 5111 1611 127 1

... Reformatted. - Wolfdieter Lang, Feb 14 2015

MAPLE

T := (n, k) -> add(binomial(n, r)*combinat:-eulerian1(r, r-k), r = k .. n):

for n from 0 to 8 do seq(T(n, k), k=0..n) od; # Peter Luschny, Jun 27 2018

MATHEMATICA

t[_, 1] = 1; t[n_, n_] = 1; t[n_, 2] = 2^(n-1)-1;

t[n_, k_] = Sum[((i-k+1)^i*(k-i)^(n-i-1) - (i-k+2)^i*(k-i-1)^(n-i-1))*Binomial[n-1, i], {i, 0, k-1}];

T[n_, k_] := t[n+1, k+1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

(* Jean-François Alcover, Jan 22 2015, after Tom Copeland *)

T[ n_, k_] := Coefficient[n! SeriesCoefficient[(1-x) Exp[t] / (1 - x Exp[(1-x) t]), {t, 0, n}] // Simplify, x, k];

Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Michael Somos, Jan 22 2015 *)

CROSSREFS

Row sums give A000522.

Cf. A008292, A123125, A248727, A074909, A007318, A000225, A066810, A028246, A001263, A119879, A001586, A019538, A090582, A123125, A130850.

Cf. A122045, A007047, A000522, A026898, A036919.

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Sep 20 2003

First formula corrected by Wolfdieter Lang, Feb 14 2015

Offset set to 0 and edited by Peter Luschny, Apr 30 2021

STATUS

approved

A349264

Generalized Euler numbers, a(n) = n!*[x^n](sec(4*x)*(sin(4*x) + 1)).

+10
23

1, 4, 16, 128, 1280, 16384, 249856, 4456448, 90767360, 2080374784, 52975108096, 1483911200768, 45344872202240, 1501108249821184, 53515555843342336, 2044143848640217088, 83285910482761809920, 3605459138582973251584, 165262072909347030040576, 7995891855149741436305408

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..19.

William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia, The generating function for the Dirichlet series Lm(s), Mathematics of Computation, Vol. 81, No. 278, pp. 1005-1023, April 2012.

Ruth Lawrence and Don Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999), no. 1, 93-107.

D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967) 689-694.

D. Shanks, Corrigendum: Generalized Euler and class numbers, Math. Comp. 22, (1968) 699.

D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

EXAMPLE

Exponential generating functions of generalized Euler numbers in context:

egf1 = sec(1*x)*(sin(x) + 1).

[A000111, A000364, A000182]

egf2 = sec(2*x)*(sin(x) + cos(x)).

[A001586, A000281, A000464]

egf3 = sec(3*x)*(sin(2*x) + cos(x)).

[A007289, A000436, A000191]

egf4 = sec(4*x)*(sin(4*x) + 1).

[A349264, A000490, A000318]

egf5 = sec(5*x)*(sin(x) + sin(3*x) + cos(2*x) + cos(4*x)).

[A349265, A000187, A000320]

egf6 = sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)).

[A001587, A000192, A000411]

egf7 = sec(7*x)*(-sin(2*x) + sin(4*x) + sin(6*x) + cos(x) + cos(3*x) - cos(5*x)).

[A349266, A064068, A064072]

egf8 = sec(8*x)*2*(sin(4*x) + cos(4*x)).

[A349267, A064069, A064073]

egf9 = sec(9*x)*(4*sin(3*x) + 2)*cos(3*x)^2.

[A349268, A064070, A064074]

MAPLE

sec(4*x)*(sin(4*x) + 1): series(%, x, 20): seq(n!*coeff(%, x, n), n = 0..19);

MATHEMATICA

m = 19; CoefficientList[Series[Sec[4*x] * (Sin[4*x] + 1), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 20 2021 *)

PROG

(PARI) seq(n)={my(x='x + O('x^(n+1))); Vec(serlaplace((sin(4*x) + 1)/cos(4*x)))} \\ Andrew Howroyd, Nov 20 2021

CROSSREFS

Cf. A000111, A001586, A007289, A349265, A001587, A349266, A349267, A349268.

Secant bisections: A000364, A000281, A000436, A000490, A000187, A000192, A064068, A064069, A064070.

Tangent bisections: A000182, A000464, A000191, A000318, A000320, A000411, A064072, A064073, A064074.

Cf. A225147, A349429.

KEYWORD

nonn

AUTHOR

Peter Luschny, Nov 20 2021

STATUS

approved

A000464

Expansion of sin x /cos 2x.
(Formerly M4812 N2059)

+10
21

1, 11, 361, 24611, 2873041, 512343611, 129570724921, 44110959165011, 19450718635716001, 10784052561125704811, 7342627959965776406281, 6023130568334172003579011, 5858598896811701995459355761, 6667317162352419006959182803611, 8776621742176931117228228227924441

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

From Peter Bala, Dec 22 2021: (Start)

Conjectures:

1) Taking the sequence (a(n))n>=1 modulo an integer k gives a purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 21 begins [11, 4, 20, 10, 17, 1, 11, 4, 20, 10, 17, 1, 11, 4, 20, 10, 17, 1, ...] with an apparent period of length 6, which divides phi(21) = 12.

2) For i >= 0, define a_i(n) = a(n+i). Then for each i the Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k. If true, then for each i the expansion of exp(Sum_{n >= 1} a_i(n)*x^n/n) has integer coefficients.

3) a(m*n) == a(m)^n (mod 2^k) for k = 2*v_2(m) + 4, where v_p(i) denotes the p-adic valuation of i.

4)(i) a(2*m*n) == a(n)^(2*m) (mod 2^k) for k = v_2(m) + 4

(ii) a((2*m+1)*n) == a(n)^(2*m+1) (mod 2^k) for k = v_2(m) + 4. (End)

REFERENCES

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.

J. W. L. Glaisher, "On the coefficients in the expansions of cos x/ cos 2x and sin x/ cos 2x", Quart. J. Pure and Applied Math., 45 (1914), 187-222.

I. J. Schwatt, Intro. to Operations with Series, Chelsea, p. 278.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..50

G. E. Andrews, J. Jimenez-Urroz, and K. Ono, q-series identities and values of certain L-functions, Duke Math Jour., Volume 108, No.3 (2001), 395-419.

Peter Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)

D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.

Simon Plouffe, Table of n, a(n) for n=0..1023

D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.

D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699

D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Alan D. Sokal, The Euler and Springer numbers as moment sequences, arXiv:1804.04498 [math.CO], 2018.

A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011.

FORMULA

E.g.f.: Sum_{k>=0} a(k)x^(2k+1)/(2k+1)! = sin(x)/cos(2x).

a(n) = (-1)^n*L(X,-2n+1) where L(X,z) is the Dirichlet L-function L(X,z) = Sum_{k>=0} X(k)/k^z and where X(k) is the Dirichlet character Legendre(k,2) which begins 1,0,-1,0,-1,0,1,0,1,0,-1,0,-1,0,1,0,1,0,-1,0.... - Benoit Cloitre, Mar 22 2009 [This Dirichlet character is A091337. - Jianing Song, Oct 22 2023]

From Peter Bala, Mar 24 2009: (Start)

Basic hypergeometric generating function:

2*exp(-t)*Sum_{n = 0..inf} (Product_{k = 1..n} (1-exp(-16*k*t))/Product_{k = 1..n+1} (1+exp(-(16*k-8)*t))) = 1 + 11*t + 361*t^2/2! + 24611*t^3/3! + .... For other sequences with generating functions of a similar type see A000364, A002105, A002439, A079144 and A158690.

a(n) = (-1)^(n+1)*L(-2*n-1), where L(s) is the Dirichlet L-function L(s) = 1 - 1/3^s - 1/5^s + 1/7^s + - - + ... [Andrews et al., Theorem 5]. (End)

From Peter Bala, Jun 18 2009: (Start)

a(n) = (-1)^n*B_(2*n+2)(X)/(2*n+2), where B_n(X) denotes the X-Bernoulli number with X a Dirichlet character modulus 8 given by X(8*n+1) = X(8*n+7) = 1, X(8*n+3) = X(8*n+5) = -1 and X(2*n) = 0. See A161722 for the values of B_n(X).

For the theory and properties of the generalized Bernoulli numbers B_n(X) and the associated generalized Bernoulli polynomials B_n(X,x) see [Cohen, Section 9.4].

The present sequence also occurs in the evaluation of the finite sum of powers Sum_{i = 0..m-1} {(8*i+1)^n - (8*i+3)^n - (8*i+5)^n + (8*i+7)^n}, n = 1,2,... - see A151751 for details. (End)

G.f. 1/G(0) where G(k) = 1 + x - x*(4*k+3)*(4*k+4)/(1 - (4*k+4)*(4*k+5)*x/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 11 2012

G.f.: 1/E(0) where E(k) = 1 - 11*x - 32*x*k*(k+1) - 16*x^2*(k+1)^2*(4*k+3)*(4*k+5)/E(k+1) (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 17 2012

a(n) ~ (2*n+1)! * 2^(4*n+7/2) / Pi^(2*n+2). - Vaclav Kotesovec, May 03 2014

a(n) = (-1)^n*2^(6*n+4)*(Zeta(-2*n-1,5/8)-Zeta(-2*n-1,7/8)). - Peter Luschny, Oct 15 2015

From Peter Bala, May 11 2017: (Start)

G.f. A(x) = 1 + 11*x + 361*x^2 + ... = 1/(1 + x - 12*x/(1 - 20*x/(1 + x - 56*x/(1 - 72*x/(1 + x - ... - 4*n*(4*n - 1)*x/(1 - 4*n*(4*n + 1)*x/((1 + x) - ...))))))).

A(x) = 1/(1 + 9*x - 20*x/(1 - 12*x/(1 + 9*x - 72*x/(1 - 56*x/(1 + 9*x - ... - 4*n*(4*n + 1)*x/(1 - 4*n*(4*n - 1)*x/(1 + 9*x - ...))))))).

It follows that the first binomial transform of A(x) and the ninth binomial transform of A(x) have continued fractions of Stieltjes-type (S-fractions). (End)

a(n) = (-1)^(n+1)*4^(2*n+1)*E(2*n+1,1/4), where E(n,x) is the n-th Euler polynomial. Cf. A002439. - Peter Bala, Aug 13 2017

From Peter Bala, Dec 04 2021: (Start)

F(x) = exp(x)*(exp(2*x) - 1)/(exp(4*x) + 1) = x - 11*x^3/3! + 361x^5/5! - 24611*x^7/7! + ... is the e.g.f. for the sequence [1, 0, -11, 0, 361, 0, -24611, 0, ...], a signed and aerated version of this sequence.

The binomial transform exp(x)*F(x) = x + 2*x^2/2! - 8*x^3/3! - 40*x^4/4! + + - - is an e.g.f. for a signed version of A000828 (omitting the initial term). (End)

From Peter Bala, Dec 22 2021: (Start)

a(1) = 1, a(n) = (-1)^(n-1) - Sum_{k = 1..n} (-4)^k*C(2*n-1,2*k)*a(n-k).

a(n) == 1 (mod 10); a(5*n+1) == 0 mod(11);

a(n) == - 23^(n+1) (mod 108); a(n) == (7^2)*59^n (mod 144);

a(n) == 11^n (mod 240); a(n) == (11^2)*131^n (mod 360). (End)

MAPLE

a := n -> (-1)^n*2^(6*n+4)*(Zeta(0, -2*n-1, 5/8)-Zeta(0, -2*n-1, 7/8)):

seq(a(n), n=0..12); # Peter Luschny, Oct 15 2015

MATHEMATICA

With[{nn=30}, Take[CoefficientList[Series[Sin[x]/Cos[2x], {x, 0, nn}], x] Range[0, nn-1]!, {2, -1, 2}]] (* Harvey P. Dale, Mar 23 2012 *)

nmax = 15; km0 = 10; d[n_, km_] := Round[(2^(4n-1/2) (2n-1)! Sum[ JacobiSymbol[2, 2k+1]/(2k+1)^(2n), {k, 0, km}])/Pi^(2n)]; dd[km_] := dd[km] = Table[d[n, km], {n, 1, nmax}]; dd[km0]; dd[km = 2*km0]; While[dd[km] != dd[km/2, km = 2*km]]; A000464 = dd[km] (* Jean-François Alcover, Feb 08 2016 *)

PROG

(PARI) a(n)=if(n<0, 0, n+=n+1; n!*polcoeff(sin(x+x*O(x^n))/cos(2*x+x*O(x^n)), n)) /* Michael Somos, Feb 09 2006 */

CROSSREFS

Cf. A064073. Bisection of A000822, A001586.

Cf. A000364, A000828, A002105, A002439, A079144, A158690.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description, new reference, Aug 15 1995

STATUS

approved

A080635

Number of permutations on n letters without double falls and without initial falls.

+10
21

1, 1, 1, 3, 9, 39, 189, 1107, 7281, 54351, 448821, 4085883, 40533129, 435847959, 5045745069, 62594829027, 828229153761, 11644113200031, 173331882039141, 2723549731505163, 45047085512477049, 782326996336904679, 14233537708408467549, 270733989894887810547

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

A permutation w has a double fall at k if w(k) > w(k+1) > w(k+2) and has an initial fall if w(1) > w(2).

exp(x*(1-y+y^2)*D_y)*f(y)|_{y=0} = f(1-E(-x)) for any function f with a Taylor series. D_y means differentiation with respect to y and E(x) is the e.g.f. given below. For a proof of exp(x*g(y)*D_y)*f(y) = f(F^{-1}(x+F(y))) with the compositional inverse F^{-1} of F(y)=int(1/g(y),y) with F(0)=0 see, e.g., the Datolli et al. reference.

Number of increasing ordered trees on vertex set {1,2,...,n}, rooted at 1, in which all outdegrees are <= 2. - David Callan, Mar 30 2007

Number of increasing colored 1-2 trees of order n with choice of two colors for the right branches of the vertices of outdegree 2. - Wenjin Woan, May 21 2011

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..464

M. Aigner, Catalan and other numbers: a recurrent theme, Algebraic combinatorics and computer science, 347-390, Springer Italia, Milan, 2001.

Cyril Banderier et al., Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016. See series expansion p. 35.

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of increasing trees, preprint.

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.

Miklós Bóna and István Mező, Limiting probabilities for vertices of a given rank in rooted trees, arXiv:1803.05033 [math.CO], 2018.

Miklós Bóna and István Mező, Limiting Probabilities for Vertices of a Given Rank in 1-2 Trees, The Electronic Journal of Combinatorics (2019) Vol. 26, No. 3, P#3.41.

Daniel Chen, Expected Number of Dice Rolls Until an Increasing Run of Three, arXiv:2308.14635 [math.CO], 2023. See p. 17.

G. Datolli, P. L. Ottaviani, A. Torre and L. Vazquez, Evolution operator equations: integration with algebraic and finite differences methods.[...], La Rivista del Nuovo Cimento 20,2 (1997) 1-133. eq. (I.2.18).

Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.

R. Ehrenborg, Cyclically consecutive permutation avoidance, arXiv:1312.2051 [math.CO], 2013.

S. Elizalde and M. Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.

M. E. Jones and J. B. Remmel, Pattern matching in the cycle structures of permutations, Pure Math. Appl. (PU.M.A.), 22 (2011) 173-208.

Kaarel Hänni, Asymptotics of descent functions, MIT Mathematics UROP+ Papers (2020).

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.

Rupert Li, Vincular Pattern Avoidance on Cyclic Permutations, arXiv:2107.12353 [math.CO], 2021, p. 6.

Christopher Zhu, Enumerating Permutations and Rim Hooks Characterized by Double Descent Sets, arXiv:1910.12818 [math.CO], 2019.

FORMULA

E.g.f.: (1 + 1/sqrt(3) * tan(sqrt(3)/2 * x)) / (1 - 1/sqrt(3) * tan( sqrt(3)/2 * x)).

Recurrence: a(n+1) = (Sum_{k=0..n} binomial(n, k) * a(k) * a(n-k)) - a(n) + 0^n.

E.g.f.: A(x) satisfies A' = 1 - A + A^2. - Michael Somos, Oct 04 2003

E.g.f.: E(x) = (3*cos((1/2)*3^(1/2)*x) + (3^(1/2))*sin((1/2)*3^(1/2)*x))/(3*cos((1/2)*3^(1/2)*x) - (3^(1/2))* sin((1/2)*3^(1/2)*x)). See the Michael Somos comment. - Wolfdieter Lang, Sep 12 2005

O.g.f.: A(x) = 1+x/(1-x-2*x^2/(1-2*x-2*3*x^2/(1-3*x-3*4*x^2/(1-... -n*x-n*(n+1)*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006

From Peter Bala: (Start)

An alternative form of the e.g.f. for this sequence taken from [Bergeron et al.] is

(1)... (sqrt(3)/2)*tan((sqrt(3)/2)*x+Pi/6) [with constant term 1/2].

By comparing the egf for this sequence with the egf for the Eulerian numbers A008292 we can show that

(2)... a(n) = A(n,w)/(1+w)^(n-1) for n >= 1,

where w = exp(2*Pi*i/3) and {A(n,x),n>=1} = [1, 1+x, 1+4*x+x^2, 1+11*x+11*x^2+x^3,...] denotes the sequence of Eulerian polynomials. Equivalently,

(3)... a(n) = (-i*sqrt(3))^(n-1)*Sum_{k=1..n} k!*Stirling2(n,k)*(-1/2 + sqrt(3)*i/6)^(k-1) for n >= 1, and

(4)... a(n) = (-i*sqrt(3))^(n-1)*Sum_{k=1..n} (-1/2 + sqrt(3)*i/6)^(k-1)* Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*j^n for n >= 1.

This explicit formula for a(n) may be used to obtain various congruence results. For example,

(5a)... a(p) == 1 (mod p) for prime p = 6*n+1,

(5b)... a(p) == -1 (mod p) for prime p = 6*n+5.

For the corresponding results for the case of non-plane unary-binary trees see A000111. For type B results see A001586. For a related sequence of polynomials see A185415. See also A185416 for a recursive method to compute this sequence. For forests of plane increasing unary binary trees see A185422 and A185423. (End)

O.g.f.: A(x) = x - (1/2)*x^2 + (1/2)*x^3 - (3/8)*x^4 + (13/40)*x^5 - (21/80)*x^6 + (123/560)*x^7 - (809/4480)*x^8 + (671/4480)*x^9 - (5541/44800)*x^10 + .... - Vladimir Kruchinin, Jan 18 2011

Let f(x) = 1+x+x^2. Then a(n+1) = (f(x)*d/dx)^n f(x) evaluated at x = 0. - Peter Bala, Oct 06 2011

From Sergei N. Gladkovskii, May 06 2013 - Dec 24 2013: (Start)

Continued fractions:

G.f.: 1 + 1/Q(0), where Q(k) = 1/(x*(k+1)) - 1 - 1/Q(k+1).

E.g.f.: 1 + 2*x/(W(0)-x), where W(k) = 4*k + 2 - 3*x^2/W(k+1).

G.f.: 1 + x/Q(0), m=1, where Q(k) = 1 - m*x*(2*k+1) - m*x^2*(2*k+1)*(2*k+2)/( 1 - m*x*(2*k+2) - m*x^2*(2*k+2)*(2*k+3)/Q(k+1) ).

G.f.: 1 + x/Q(0), where Q(k) = 1 - x*(k+1) - x^2*(k+1)*(k+2)/Q(k+1).

G.f.: 1 + T(0)*x/(1-x), where T(k) = 1 - x^2*(k+1)*(k+2)/( x^2*(k+1)*(k+2) - (1-x*(k+1))*(1-x*(k+2))/T(k+1) ).

G.f.: 1 + x/(G(0)-x), where G(k) = 1 + x*(k+1) - x*(k+1)/(1 - x*(k+2)/G(k+1) ). (End)

a(n) ~ 3^(3*(n+1)/2) * n^(n+1/2) / (exp(n)*(2*Pi)^(n+1/2)). - Vaclav Kotesovec, Oct 05 2013

a(n) = n! * Sum_{k=-oo..oo} (sqrt(3)/(2*Pi*(k+1/3)))^(n+1) for n >= 1. - Richard Ehrenborg, Dec 09 2013

From Peter Bala, Sep 11 2015: (Start)

The e.g.f. A(x) = (sqrt(3)/2)*tan((sqrt(3)/2)*x + Pi/6) satisfies the differential equation A"(x) = 2*A(x)*A'(x) with A(0) = 1/2 and A'(0) = 1, leading to the recurrence a(0) = 1/2, a(1) = 1, else a(n) = 2*Sum_{i = 0..n-2} binomial(n-2,i)*a(i)*a(n-1-i) for the sequence [1/2, 1, 1, 3, 9, 39, 189, 1107, ...].

Note, the same recurrence, but with the initial conditions a(0) = 1 and a(1) = 1, produces the sequence n! and with a(0) = 0 and a(1) = 1 produces A000182. Cf. A002105, A234797. (End)

E.g.f.: exp( Series_Reversion( Integral 1/(2*cosh(x) - 1) dx ) ). - Paul D. Hanna, Feb 22 2016

a(n) = Sum_{k=1..n} (-3)^((n-k)/2)*((-1)^(n-k)+1)*Sum_{j=0..n-k} C(j+k-1,j)*(j+k)!*2^(-j-k)*(-1)^j*Stirling2(n,j+k),n>0, a(0)=1. - Vladimir Kruchinin, Feb 13 2019

For n > 0, a(n) = 3^((n+1)/2) * n! * (zeta(n+1, 1/3) - (-1)^n*zeta(n+1, 2/3)) / (2*Pi)^(n+1). - Vaclav Kotesovec, Aug 06 2021

EXAMPLE

E.g.f. = 1 + x + (1/2)*x^2 + (1/2)*x^3 + (3/8)*x^4 + (13/40)*x^5 + (21/80)*x^6 + ...

G.f. = 1 + x + x^2 + 3*x^3 + 9*x^4 + 39*x^5 + 189*x^6 + 1107*x^7 + ...

For n = 3: 123, 132, 231. For n = 4: 1234, 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.

a(4)=9. The 9 plane (ordered) increasing unary-binary trees are

...................................................................

..4................................................................

..|................................................................

..3..........4...4...............4...4...............3...3.........

..|........./.....\............./.....\............./.....\........

..2....2...3.......3...2...3...2.......2...3...4...2.......2...4...

..|.....\./.........\./.....\./.........\./.....\./.........\./....

..1......1...........1.......1...........1.......1...........1.....

...................................................................

..3...4...4...3....................................................

...\./.....\./.....................................................

....2.......2......................................................

....|.......|......................................................

....1.......1......................................................

...................................................................

MAPLE

a:= proc(n) if n < 2 then 1 else n! * sum((sqrt(3)/(2*Pi*(k+1/3)))^(n+1), k=-infinity..infinity) fi end: seq(a(n), n=0..30); # Richard Ehrenborg, Dec 09 2013

a := proc(n) option remember; local k; if n < 3 then 1 else

add(binomial(n-1, k)*a(k)*a(n-k-1), k = 0..n-2) fi end:

seq(a(n), n = 0..23); # Peter Luschny, May 24 2024

MATHEMATICA

Table[n!, {n, 0, 40}]*CoefficientList[Series[ (1 + 1/Sqrt[3] Tan[Sqrt[3]/2 x])/(1 - 1/Sqrt[3] Tan[Sqrt[3]/2 x]), {x, 0, 40}], x]

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1/2 + Sqrt[3]/2 Tan[ Pi/6 + Sqrt[3] x/2], {x, 0, n}]]; (* Michael Somos, May 22 2011 *)

Join[{1}, FullSimplify[Table[3^((n+1)/2) * n! * (Zeta[n+1, 1/3] - (-1)^n*Zeta[n+1, 2/3]) / (2*Pi)^(n+1), {n, 1, 20}]]] (* Vaclav Kotesovec, Aug 06 2021 *)

PROG

(PARI) {a(n) = my(A); if( n<1, n==0, A = O(x); for( k=1, n, A = intformal( 1 + A + A^2)); n! * polcoeff( A, n))}; /* Michael Somos, Oct 04 2003 */

(Sage)

@CachedFunction

def c(n, k) :

if n==k: return 1

if k<1 or k>n: return 0

return ((n-k)//2+1)*c(n-1, k-1)+2*k*c(n-1, k+1)

def A080635(n):

return add(c(n, k) for k in (0..n))

[A080635(n) for n in (0..23)] # Peter Luschny, Jun 10 2014

(PARI) {a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(2*cosh(x +x*O(x^n)) - 1) ) )), n)}

for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 22 2016

(Maxima)

a(n):=if n=0 then 1 else sum((-3)^((n-k)/2)*((-1)^(n-k)+1)*sum(binomial(j+k-1, j)*(j+k)!*2^(-j-k)*(-1)^(j)*stirling2(n, j+k), j, 0, n-k), k, 1, n); /* Vladimir Kruchinin, Feb 13 2019 */

CROSSREFS

Cf. A049774, A185415, A185416, A185422, A185423, A139605, A232864.

Cf. A000182, A002105, A234797.

Column k=0 of A162976.

KEYWORD

nonn,easy

AUTHOR

Emanuele Munarini, Feb 28 2003

EXTENSIONS

Several typos corrected by Olivier Gérard, Mar 26 2011

STATUS

approved

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