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%I #20 May 20 2021 04:44:52
%S 1,1,0,1,-2,0,1,-3,4,0,1,-4,6,-8,0,1,-5,10,-9,16,0,1,-6,15,-20,9,-32,
%T 0,1,-7,21,-35,36,0,64,0,1,-8,28,-56,70,-64,-27,-128,0,1,-9,36,-84,
%U 126,-125,120,81,256,0,1,-10,45,-120,210,-252,200,-240,-162,-512,0
%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="/A307047/b307047.txt">Antidiagonals n = 0..139, flattened</a>
%F A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^((k 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 0, -2, -3, -4, -5, -6, -7, -8, ...
%e 0, 4, 6, 10, 15, 21, 28, 36, ...
%e 0, -8, -9, -20, -35, -56, -84, -120, ...
%e 0, 16, 9, 36, 70, 126, 210, 330, ...
%e 0, -32, 0, -64, -125, -252, -462, -792, ...
%e 0, 64, -27, 120, 200, 463, 924, 1716, ...
%e 0, -128, 81, -240, -275, -804, -1715, -3432, ...
%e 0, 256, -162, 496, 275, 1365, 2989, 6436, ...
%t T[n_, k_] := (-1)^n * Sum[(-1)^(j * Mod[k, 2]) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 20 2021 *)
%Y Columns 1-7 give A000007, A122803, A000748, A000749(n+3), A000750, A006090, A049018.
%Y Cf. A039912 (square array A(n,k), n >= 0, k >= 2), A306913, A306914, A306915.
%K sign,tabl
%O 0,5
%A _Seiichi Manyama_, Mar 21 2019
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