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A307047
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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).
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4
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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, 0, 1, -7, 21, -35, 36, 0, 64, 0, 1, -8, 28, -56, 70, -64, -27, -128, 0, 1, -9, 36, -84, 126, -125, 120, 81, 256, 0, 1, -10, 45, -120, 210, -252, 200, -240, -162, -512, 0
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OFFSET
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0,5
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LINKS
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FORMULA
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A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^((k mod 2) * j) * binomial(n+k-1,k*j+k-1).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, -2, -3, -4, -5, -6, -7, -8, ...
0, 4, 6, 10, 15, 21, 28, 36, ...
0, -8, -9, -20, -35, -56, -84, -120, ...
0, 16, 9, 36, 70, 126, 210, 330, ...
0, -32, 0, -64, -125, -252, -462, -792, ...
0, 64, -27, 120, 200, 463, 924, 1716, ...
0, -128, 81, -240, -275, -804, -1715, -3432, ...
0, 256, -162, 496, 275, 1365, 2989, 6436, ...
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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