A199899 -id:A199899

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A193844

Triangular array: the fission of ((x+1)^n) by ((x+1)^n); i.e., the self-fission of Pascal's triangle.

+10
6

1, 1, 3, 1, 5, 7, 1, 7, 17, 15, 1, 9, 31, 49, 31, 1, 11, 49, 111, 129, 63, 1, 13, 71, 209, 351, 321, 127, 1, 15, 97, 351, 769, 1023, 769, 255, 1, 17, 127, 545, 1471, 2561, 2815, 1793, 511, 1, 19, 161, 799, 2561, 5503, 7937, 7423, 4097, 1023

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

OFFSET

0,3

COMMENTS

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

A193844 is also the fission of (p1(n,x)) by (q1(n,x)), where p1(n,x)=x^n+x^(n-1)+...+x+1 and q1(n,x)=(x+2)^n.

Essentially A119258 but without the main diagonal. - Peter Bala, Jul 16 2013

From Robert Coquereaux, Oct 02 2014: (Start)

This is also a rectangular array A(n,p) read down the antidiagonals:

1 1 1 1 1 1 1 1 1

3 5 7 9 11 13 15 17 19

7 17 31 49 71 97 127 161 199

15 49 111 209 351 545 799 1121 1519

31 129 351 769 1471 2561 4159 6401 9439

...

Calling Gr(n) the Grassmann algebra with n generators, A(n,p) is the dimension of the space of Gr(n)-valued symmetric multilinear forms with vanishing graded divergence. If p is odd A(n,p) is the dimension of the cyclic cohomology group of order p of the Z2 graded algebra Gr(n). If p is even, the dimension of this cohomology group is A(n,p)+1. A(n,p) = 2^n*A059260(p,n-1)-(-1)^p.

(End)

The n-th row are also the coefficients of the polynomial P=sum_{k=0..n} (X+2)^k (in falling order, i.e., that of X^n first). - M. F. Hasler, Oct 15 2014

LINKS

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

Jean-François Chamayou, A Random Difference Equation with Dufresne Variables revisited, arXiv:1410.1708 [math.PR], 2014.

R. Coquereaux and E. Ragoucy, Currents on Grassmann algebras, J. of Geometry and Physics, 1995, Vol 15, pp 333-352.

R. Coquereaux and E. Ragoucy, Currents on Grassmann algebras, arXiv:hep-th/9310147, 1993.

C. Kassel, A Künneth formula for the cyclic cohomology of Z2-graded algebras, Math. Ann. 275 (1986) 683.

FORMULA

From Peter Bala, Jul 16 2013: (Start)

T(n,k) = sum {i = 0..k} (-1)^i*binomial(n+1,k-i)*2^(k-i).

O.g.f.: 1/( (1 - x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + 3*x)*t + (1 + 5*x + 7*x^2)*t^2 + ....

The n-th row polynomial R(n,x) = 1/(x+1)*( (2*x+1)^(n+1) - x^(n+1) ). (End)

T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - T(n-2,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 17 2014

T(n,k) = 2^k*binomial(n+1,k)*hyper2F1(1,-k,-k+n+2, 1/2). - Peter Luschny, Jul 23 2014

EXAMPLE

First six rows:

1....3

1....5....7

1....7....17....15

1....9....31....49....31

1....11...49....111...129...63

MAPLE

A193844 := (n, k) -> 2^k*binomial(n+1, k)*hypergeom([1, -k], [-k+n+2], 1/2);

for n from 0 to 5 do seq(round(evalf(A193844(n, k))), k=0..n) od; # Peter Luschny, Jul 23 2014

# Alternatively

p := (n, x) -> add(x^k*(1+2*x)^(n-k), k=0..n): for n from 0 to 7 do [n], PolynomialTools:-CoefficientList(p(n, x), x) od; # Peter Luschny, Jun 18 2017

MATHEMATICA

z = 10;

p[n_, x_] := (x + 1)^n;

q[n_, x_] := (x + 1)^n

p1[n_, k_] := Coefficient[p[n, x], x^k];

p1[n_, 0] := p[n, x] /. x -> 0;

d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]

h[n_] := CoefficientList[d[n, x], {x}]

TableForm[Table[Reverse[h[n]], {n, 0, z}]]

Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A193844 *)

TableForm[Table[h[n], {n, 0, z}]]

Flatten[Table[h[n], {n, -1, z}]] (* A193845 *)

PROG

(Sage) # uses[fission from A193842]

p = lambda n, x: (x+1)^n

A193844_row = lambda n: fission(p, p, n)

for n in range(7): print(A193844_row(n)) # Peter Luschny, Jul 23 2014

CROSSREFS

Cf. A193842, A193845, A119258.

Columns, diagonals: A000225, A000337, A055580, A027608, A211386, A211388, A000012, A005408, A056220, A199899.

A145661 is an essentially identical triangle.

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 07 2011

STATUS

approved

A193845

Mirror of the triangle A193844.

+10
4

1, 3, 1, 7, 5, 1, 15, 17, 7, 1, 31, 49, 31, 9, 1, 63, 129, 111, 49, 11, 1, 127, 321, 351, 209, 71, 13, 1, 255, 769, 1023, 769, 351, 97, 15, 1, 511, 1793, 2815, 2561, 1471, 545, 127, 17, 1, 1023, 4097, 7423, 7937, 5503, 2561, 799, 161, 19, 1

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

OFFSET

0,2

COMMENTS

This triangle is obtained by reversing the rows of the triangle A193844.

From Philippe Deléham, Jan 17 2014: (Start)

Subtriangle of the triangle in A112857.

T(n,0) = A000225(n+1).

T(n,1) = A000337(n).

T(n+2,2) = A055580(n).

T(n+3,3) = A027608(n).

T(n+4,4) = A211386(n).

T(n+5,5) = A211388(n).

T(n,n) = A000012(n).

T(n+1,n) = A005408(n).

T(n+2,n) = A056220(n+2).

T(n+3,n) = A199899(n+1).

Row sums are A003462(n+1).

Diagonal sums are A048739(n).

Riordan array (1/((1-2*x)*(1-x), x/(1-2*x)). (End)

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-2)^0 + A_1*(x-2)^1 + A_2*(x-2)^2 + ... + A_n*(x-2)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0. - Derek Orr, Oct 14 2014

The n-th row lists the coefficients of the polynomial sum_{k=0..n} (X+2)^k, in order of increasing powers. - M. F. Hasler, Oct 15 2014

LINKS

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

Russell Jay Hendel, A Method for Uniformly Proving a Family of Identities, arXiv:2107.03549 [math.CO], 2021.

FORMULA

T(n,k) = A193844(n,n-k).

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 17 2014

EXAMPLE

First six rows:

3....1

7....5....1

15...17...7....1

31...49...31...9...1

63...129..111..49..11..1