A180563 -id:A180563

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A294332

G.f.: exp( Sum_{n>=1} A180563(n) * x^n / n ).

+20
2

1, 1, -1, 5, -45, 609, -11141, 257281, -7170355, 233936995, -8744103079, 368479396171, -17288353555771, 894005702731735, -50527305282004435, 3099060459670425655, -205028564671300495120, 14554510561318327509610, -1103542106915790217739110, 89009707681627448130203830, -7610129271299704960998906454, 687495658528174987634449288846, -65438091790081511530153327883206, 6545685493719560524729653911676430

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

OFFSET

0,4

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

EXAMPLE

G.f.: A(x) = 1 + x - x^2 + 5*x^3 - 45*x^4 + 609*x^5 - 11141*x^6 + 257281*x^7 - 7170355*x^8 + 233936995*x^9 - 8744103079*x^10 +...

such that

log(A(x)) = x - 3*x^2/2 + 19*x^3/3 - 207*x^4/4 + 3331*x^5/5 - 71223*x^6/6 + 1890379*x^7/7 - 59652687*x^8/8 + 2175761971*x^9/9 +...+ A180563(n)*x^n/n +...

where the e.g.f. G(x) of A180563 begins

G(x) = x - 3*x^2/2! + 19*x^3/3! - 207*x^4/4! + 3331*x^5/5! - 71223*x^6/6! + 1890379*x^7/7! +...+ A180563(n)*x^n/n! +...

and satisfies: Product_{n>=1} (1 - G(x)^n) = exp(-x).

PROG

(PARI) {A180563(n) = my( L = sum(m=1, n, sigma(m) * x^m/m ) +x*O(x^n) ); n!*polcoeff( serreverse(L), n)}

{a(n) = my(A); A = exp( sum(m=1, n+1, A180563(m)*x^m/m +x*O(x^n)) ); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A180653, A294331 (variant).

KEYWORD

sign

AUTHOR

Paul D. Hanna, Oct 28 2017

STATUS

approved

A008298

Triangle of D'Arcais numbers.

+10
12

1, 3, 1, 8, 9, 1, 42, 59, 18, 1, 144, 450, 215, 30, 1, 1440, 3394, 2475, 565, 45, 1, 5760, 30912, 28294, 9345, 1225, 63, 1, 75600, 293292, 340116, 147889, 27720, 2338, 84, 1, 524160, 3032208, 4335596, 2341332, 579369, 69552, 4074, 108, 1, 6531840, 36290736, 57773700, 38049920, 11744775, 1857513, 154350, 6630, 135, 1

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

OFFSET

1,2

COMMENTS

Also the Bell transform of A038048(n+1) and the inverse Bell transform of A180563(n+1) (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Named after the Italian mathematician Francesco Flores D'Arcais (1849-1927). - Amiram Eldar, Jun 13 2021

REFERENCES

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.

F. D'Arcais, Développement en série, Intermédiaire Math., Vol. 20 (1913), pp. 233-234.

LINKS

Seiichi Manyama, Rows n = 1..100, flattened (rows n = 1..20 from Vincenzo Librandi)

Peter Luschny, The Bell transform.

FORMULA

G.f.: Sum_{1<=k<=n} T(n, k)*u^k*t^n/n! = ((1-t)*(1-t^2)*(1-t^3)...)^(-u).

Recurrence for degree n D'Arcais polynomials T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = Sum_{k=1..n} (n-1)!/(n-k)!*sigma(k)*u*T(n-k; u), T(0; u) = 1. - Vladeta Jovovic, Oct 11 2002

T(n; u) = n!*Sum_{pi} Product_{i=1..n} binomial(u+k(i)-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic, Oct 11 2002

E.g.f.: exp(Sum_{n>0} sigma(n)*u*x^n/n), where sigma(n)=A000203(n). - Vladeta Jovovic, Jan 10 2003

T(n, k) = coeff(n!*P(n), x^k), n >= 1 and 1 <= k <= n, with P(n) = (1/n)*Sum_{k=0..n-1} sigma(n-k)*P(k)*x for n >= 1 and P(n=0) = 1. See A036039. - Johannes W. Meijer, Jul 08 2016

T(n, k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} sigma(i_j)/i_j. - Seiichi Manyama, Nov 09 2020.

EXAMPLE

exp(Sum_{n>0} sigma(n)*u*x^n/n) = 1+u*x/1!+(3*u+u^2)*x^2/2!+(8*u+9*u^2+u^3)*x^3/3!+(42*u+59*u^2+18*u^3+u^4)*x^4/4!+...

Triangle starts:

3, 1;

8, 9, 1;

42, 59, 18, 1;

144, 450, 215, 30, 1;

1440, 3394, 2475, 565, 45, 1;

5760, 30912, 28294, 9345, 1225, 63, 1;

75600, 293292, 340116, 147889, 27720, 2338, 84, 1;

...

T(4; u) = 4!*(binomial(u+3,4) + binomial(u+1,2)*binomial(u,1) + binomial(u+1,2) + binomial(u,1)^2 + binomial(u,1)) = 42*u+59*u^2+18*u^3+u^4.

MAPLE

P := proc(n): if n=0 then 1 else P(n):= (1/n)*(add(x(n-k) * P(k), k=0..n-1)) fi; end: with(numtheory): x := proc(n): sigma(n) * x end: Q := proc(n): n!*P(n) end: T := proc(n, k): coeff(Q(n), x, k) end: seq(seq(T(n, k), k=1..n), n=1..10); # Johannes W. Meijer, Jul 08 2016

MATHEMATICA

t[0][u_] = 1; t[n_][u_] := t[n][u] = Sum[(n-1)!/(n-k)!*DivisorSigma[1, k]*u*t[n-k][u], {k, 1, n}]; row[n_] := CoefficientList[ t[n][u], u] // Rest; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Oct 03 2012, after Vladeta Jovovic *)

PROG

(Sage) # uses[bell_matrix from A264428]

# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.

print(bell_matrix(lambda n: A038048(n+1), 9)) # Peter Luschny, Jan 19 2016

(PARI) row(n)={local(P(n)=if(n, sum(k=0, n-1, sigma(n-k)*x*P(k))/n, 1)); Vecrev(P(n)*n!/x)} \\ T(n, k)=row(n)[k]. - M. F. Hasler, Jul 13 2016

(PARI) a(n) = if(n<1, 0, (n-1)!*sigma(n));

T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1))) \\ Seiichi Manyama, Nov 08 2020 after Peter Luschny

CROSSREFS

Column k=1..3 give A038048, A059356, A059357.

Row sums give A053529.

Cf. A075525, A180563, A210590.

KEYWORD

nonn,tabl,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Dec 28 2001

STATUS

approved

A294330

E.g.f. A(x) satisfies: Product_{n>=1} (1 - (-A(x))^n) = exp(x).

+10
3

1, 3, 19, 207, 3331, 71223, 1890379, 59652687, 2175761971, 89953773543, 4155502117339, 212122704251967, 11857607972675011, 720435277883199063, 47273215180877201899, 3331797538738820992047, 251025685429022007354451, 20133640365773761748643783, 1712740622904757368673592059

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

OFFSET

1,2

COMMENTS

Unsigned version of A180563.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..300

FORMULA

E.g.f. A(x) satisfies:

(1) Sum_{n>=1} (-1)^(n-1) * sigma(n) * A(x)^n / n = x.

(2) Sum_{n>=0} (-1)^[n/2] * (2*n+1) * A(x)^(n*(n+1)/2) = exp(3*x).

(3) A(x) = Series_Reversion( log(Q(x)) ) where Q(x) = Product_{n>=1} (1 - (-x)^n).

a(n) ~ c * d^n * n^(n-1), where d = 1.788680223969315995... and c = 0.254472375755339325... - Vaclav Kotesovec, Oct 29 2017

EXAMPLE

E.g.f.: A(x) = x + 3*x^2/2! + 19*x^3/3! + 207*x^4/4! + 3331*x^5/5! + 71223*x^6/6! + 1890379*x^7/7! + 59652687*x^8/8! + 2175761971*x^9/9! + 89953773543*x^10/10! +...

such that A( log(Q(x)) ) = x, where:

Q(x) = Product_{n>=1} (1 - (-x)^n);

log(Q(x)) = x - 3*x^2/2 + 4*x^3/3 - 7*x^4/4 + 6*x^5/5 - 12*x^6/6 + 8*x^7/7 - 15*x^8/8 + 13*x^9/9 - 18*x^10/10 +...+ (-1)^(n-1)*sigma(n)*x^n/n +...

and Q(x) = 1 + x - x^2 - x^5 - x^7 - x^12 + x^15 + x^22 + x^26 + x^35 - x^40 - x^51 - x^57 - x^70 + x^77 + x^92 + x^100 +...+ A121373(n)*x^n +...

Also,

exp(3*x) = 1 + 3*A(x) - 5*A(x)^3 - 7*A(x)^6 + 9*A(x)^10 + 11*A(x)^15 - 13*A(x)^21 - 15*A(x)^28 + 17*A(x)^36 +...+ (-1)^[n/2] * (2*n+1) * A(x)^(n*(n+1)/2) +...

ALTERNATE GENERATING FUNCTION.

L.g.f.: L(x) = x + 3*x^2/2 + 19*x^3/3 + 207*x^4/4 + 3331*x^5/5 + 71223*x^6/6 + 1890379*x^7/7 + 59652687*x^8/8 + 2175761971*x^9/9 + 89953773543*x^10/10 +...

such that

exp(L(x)) = 1 + x + 2*x^2 + 8*x^3 + 60*x^4 + 732*x^5 + 12672*x^6 + 283704*x^7 + 7757526*x^8 + 249885110*x^9 + 9255184676*x^10 +...+ A294331(n)*x^n +...

MATHEMATICA

(* Calculation of constants {d, c}: *) eq = FindRoot[{E^r == QPochhammer[-s], (E^r*(Log[1 + s] + QPolyGamma[0, 1, -s]))/(s*Log[-s]) + Derivative[0, 1][QPochhammer][-s, -s] == 0}, {r, 1/5}, {s, 1/2}, WorkingPrecision -> 400]; {N[1/r/E /. eq, 120], val = s*E^r*Sqrt[-r*(1 + s) * (Log[-s]^2/(E^(2*r)*(1 + s)*QPolyGamma[1, 1, -s] + s*Log[-s]*(-s*(1 + s) * Log[-s] * Derivative[0, 1][QPochhammer][-s, -s]^2 + E^r*(1 + s)*((-2 - Log[-s]) * Derivative[0, 1][QPochhammer][-s, -s] + s*Log[-s] * Derivative[0, 2][QPochhammer][-s, -s]) + 2*E^(2*r)*(-1 + (1 + s) * Derivative[0, 0, 1][QPolyGamma][0, 1, -s]))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Sep 28 2023 *)

PROG

(PARI) {a(n) = local( L = sum(m=1, n, (-1)^(m-1) * sigma(m) * x^m/m ) +x*O(x^n) ); n!*polcoeff( serreverse(L), n)}

for(n=1, 20, print1(a(n), ", "))

CROSSREFS

Cf. A294331, A010815, A180563 (variant).

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 28 2017

STATUS

approved

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