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%I #22 May 26 2021 00:55:12
%S 1,1,-2,1,-2,4,1,-3,2,-8,1,-4,6,0,16,1,-5,10,-11,-4,-32,1,-6,15,-20,
%T 21,8,64,1,-7,21,-35,34,-42,-8,-128,1,-8,28,-56,70,-48,85,0,256,1,-9,
%U 36,-84,126,-127,48,-171,16,-512,1,-10,45,-120,210,-252,220,0,342,-32,1024
%N Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1+x)^k+x^k).
%H Seiichi Manyama, <a href="/A306913/b306913.txt">Antidiagonals n = 0..139, flattened</a>
%F A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^(((k+1) mod 2) * j) * binomial(n+k-1,k*j+k-1).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e -2, -2, -3, -4, -5, -6, -7, -8, ...
%e 4, 2, 6, 10, 15, 21, 28, 36, ...
%e -8, 0, -11, -20, -35, -56, -84, -120, ...
%e 16, -4, 21, 34, 70, 126, 210, 330, ...
%e -32, 8, -42, -48, -127, -252, -462, -792, ...
%e 64, -8, 85, 48, 220, 461, 924, 1716, ...
%e -128, 0, -171, 0, -385, -780, -1717, -3432, ...
%e 256, 16, 342, -164, 715, 1209, 3017, 6434, ...
%t A[n_, k_] := (-1)^n * Sum[(-1)^(Mod[k+1, 2] * j) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[A[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 25 2021 *)
%Y Columns 1-2 give A122803, A108520.
%Y Cf. A039912, A306914, A306915.
%K sign,tabl
%O 0,3
%A _Seiichi Manyama_, Mar 16 2019