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%I #14 May 13 2021 02:35:24
%S 1,1,1,1,2,3,1,2,5,10,1,2,6,14,35,1,2,6,19,43,126,1,2,6,20,62,142,462,
%T 1,2,6,20,69,207,494,1716,1,2,6,20,70,242,705,1780,6435,1,2,6,20,70,
%U 251,858,2445,6563,24310,1,2,6,20,70,252,912,3068,8622,24566,92378
%N A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j*binomial(2*n,k*j+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 2, 2, 2, 2, 2, ...
%e 3, 5, 6, 6, 6, 6, 6, ...
%e 10, 14, 19, 20, 20, 20, 20, ...
%e 35, 43, 62, 69, 70, 70, 70, ...
%e 126, 142, 207, 242, 251, 252, 252, ...
%e 462, 494, 705, 858, 912, 923, 924, ...
%e 1716, 1780, 2445, 3068, 3341, 3418, 3431, ...
%e 6435, 6563, 8622, 11051, 12310, 12750, 12854, ...
%t T[n_, k_] := Sum[(-1)^j*Binomial[2*n, k*j + n], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 13 2021*)
%Y Columns 1-2 give A088218, A005317.
%Y Cf. A306914, A307039, A307394, A307665.
%K nonn,tabl
%O 0,5
%A _Seiichi Manyama_, Apr 20 2019
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