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A306915
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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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8
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1, 1, 2, 1, 2, 4, 1, 3, 4, 8, 1, 4, 6, 8, 16, 1, 5, 10, 11, 16, 32, 1, 6, 15, 20, 21, 32, 64, 1, 7, 21, 35, 36, 42, 64, 128, 1, 8, 28, 56, 70, 64, 85, 128, 256, 1, 9, 36, 84, 126, 127, 120, 171, 256, 512, 1, 10, 45, 120, 210, 252, 220, 240, 342, 512, 1024
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OFFSET
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0,3
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LINKS
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FORMULA
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A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+k-1,k*j+k-1).
A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - Seiichi Manyama, Apr 07 2019
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
2, 2, 3, 4, 5, 6, 7, 8, ...
4, 4, 6, 10, 15, 21, 28, 36, ...
8, 8, 11, 20, 35, 56, 84, 120, ...
16, 16, 21, 36, 70, 126, 210, 330, ...
32, 32, 42, 64, 127, 252, 462, 792, ...
64, 64, 85, 120, 220, 463, 924, 1716, ...
128, 128, 171, 240, 385, 804, 1717, 3432, ...
256, 256, 342, 496, 715, 1365, 3017, 6436, ...
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MATHEMATICA
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A[n_, k_] := Sum[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 *)
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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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