cp's OEIS Frontend

A361834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * binomial(2*j,j) * binomial(k*j,n-j).

%I A361834 #15 Mar 26 2023 11:14:46
%S A361834 1,1,2,1,2,6,1,2,4,20,1,2,2,8,70,1,2,0,-2,16,252,1,2,-2,-10,-14,32,
%T A361834 924,1,2,-4,-16,-22,-32,64,3432,1,2,-6,-20,-10,12,-30,128,12870,1,2,
%U A361834 -8,-22,20,118,174,64,256,48620,1,2,-10,-22,66,242,304,344,346,512,184756
%N A361834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * binomial(2*j,j) * binomial(k*j,n-j).
%F A361834 G.f. of column k: 1/sqrt(1 - 4*x*(1-x)^k).
%F A361834 n*T(n,k) = 2 * Sum_{j=0..k} (-1)^j * binomial(k,j)*(2*n-1-j)*T(n-1-j,k) for n > k.
%e A361834 Square array begins:
%e A361834     1,  1,   1,   1,   1,   1,    1, ...
%e A361834     2,  2,   2,   2,   2,   2,    2, ...
%e A361834     6,  4,   2,   0,  -2,  -4,   -6, ...
%e A361834    20,  8,  -2, -10, -16, -20,  -22, ...
%e A361834    70, 16, -14, -22, -10,  20,   66, ...
%e A361834   252, 32, -32,  12, 118, 242,  342, ...
%e A361834   924, 64, -30, 174, 304,  82, -678, ...
%o A361834 (PARI) T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(2*j, j)*binomial(k*j, n-j));
%Y A361834 Columns k=0..4 give A000984, A000079, A361815, A361816, A361817.
%Y A361834 Main diagonal gives A361835.
%Y A361834 Cf. A361830.
%K A361834 sign,tabl
%O A361834 0,3
%A A361834 _Seiichi Manyama_, Mar 26 2023