A353596 - OEIS

%I #17 Oct 04 2023 04:49:14

%S 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,

%T -84,0,-40,0,280,0,-504,0,264,-10,0,280,0,-1260,0,1848,0,-858,0,140,0,

%U -1680,0,5544,0,-6864,0,2860,28,0,-1260,0,9240,0,-24024,0,25740,0,-9724

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

%e Triangle T(n, k) starts:

%e [0]   1;

%e [1]   0,   2;

%e [2]   2,   0,  -2;

%e [3]   0,  -4,   0,     4;

%e [4]  -2,   0,  12,     0,   -10;

%e [5]   0,  12,   0,   -40,     0,   28;

%e [6]   4,   0, -60,     0,   140,    0,  -84;

%e [7]   0, -40,   0,   280,     0, -504,    0,   264;

%e [8] -10,   0, 280,     0, -1260,    0, 1848,     0, -858;

%e [9]   0, 140,   0, -1680,     0, 5544,    0, -6864,    0, 2860;

%e .

%e Unsigned antidiagonals |T(n+k, n-k)|:

%e [0]  1;

%e [1]  2,   2;

%e [2]  2,   4,    2;

%e [3]  4,  12,   12,    4;

%e [4] 10,  40,   60,   40,   10;

%e [5] 28, 140,  280,  280,  140,  28;

%e [6] 84, 504, 1260, 1680, 1260, 504, 84;

%p g := n -> (-2)^n*GegenbauerC(n, -1/2, x):

%p seq(print(seq(coeff(simplify(g(n)), x, k), k = 0..n)), n = 0..9);

%t 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 *)

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

%K sign,tabl

%O 0,3

%A _Peter Luschny_, May 06 2022