A353596 Triangle read by rows, T(n, k) = [x^k] (-2)^n*GegenbauerC(n, -1/2, x).

1, 0, 2, 2, 0, -2, 0, -4, 0, 4, -2, 0, 12, 0, -10, 0, 12, 0, -40, 0, 28, 4, 0, -60, 0, 140, 0, -84, 0, -40, 0, 280, 0, -504, 0, 264, -10, 0, 280, 0, -1260, 0, 1848, 0, -858, 0, 140, 0, -1680, 0, 5544, 0, -6864, 0, 2860, 28, 0, -1260, 0, 9240, 0, -24024, 0, 25740, 0, -9724

Offset

0,3

Links

Table of n, a(n) for n=0..65.

Example

Triangle T(n, k) starts:
[0]   1;
[1]   0,   2;
[2]   2,   0,  -2;
[3]   0,  -4,   0,     4;
[4]  -2,   0,  12,     0,   -10;
[5]   0,  12,   0,   -40,     0,   28;
[6]   4,   0, -60,     0,   140,    0,  -84;
[7]   0, -40,   0,   280,     0, -504,    0,   264;
[8] -10,   0, 280,     0, -1260,    0, 1848,     0, -858;
[9]   0, 140,   0, -1680,     0, 5544,    0, -6864,    0, 2860;
.
Unsigned antidiagonals |T(n+k, n-k)|:
[0]  1;
[1]  2,   2;
[2]  2,   4,    2;
[3]  4,  12,   12,    4;
[4] 10,  40,   60,   40,   10;
[5] 28, 140,  280,  280,  140,  28;
[6] 84, 504, 1260, 1680, 1260, 504, 84;

Maple

g := n -> (-2)^n*GegenbauerC(n, -1/2, x):
seq(print(seq(coeff(simplify(g(n)), x, k), k = 0..n)), n = 0..9);

Mathematica

s={}; For[n=0, n<11, n++, For[k=0, k<n+1, k++, If[EvenQ[n+k], If[Mod[n+k, 4]==0, AppendTo[s, Binomial[n+k, (n+k)/2]/(1-(n+k))*Binomial[(n+k)/2, k]], AppendTo[s, (-1)*Binomial[n+k, (n+k)/2]/(1-(n+k))*Binomial[(n+k)/2, k]]], AppendTo[s, 0]]]]; s (* Detlef Meya, Oct 03 2023 *)

Crossrefs

Diagonals (also divided by 2^k): A002420 (main), A028329 (main-2) (also A000984), A005430 (main-4) (also A002457), A002802 (main-6).

Sequence in context: A120439 A248512 A352564 * A182122 A104624 A371711

Adjacent sequences: A353593 A353594 A353595 * A353597 A353598 A353599

Keyword

sign,tabl

Author

Peter Luschny, May 06 2022

Status

approved