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A307079
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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-x)^(k-2))/((1-x)^k+x^k).
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5
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1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 0, 1, 1, 2, 3, 3, -4, 1, 1, 2, 3, 4, 0, -8, 1, 1, 2, 3, 4, 4, -9, -8, 1, 1, 2, 3, 4, 5, 0, -27, 0, 1, 1, 2, 3, 4, 5, 5, -14, -54, 16, 1, 1, 2, 3, 4, 5, 6, 0, -48, -81, 32, 1, 1, 2, 3, 4, 5, 6, 6, -20, -116, -81, 32, 1
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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) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+1,k*j+1).
A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i,k*j) * binomial(n-i,k*j).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, 2, 2, 2, ...
1, 2, 3, 3, 3, 3, 3, 3, 3, ...
1, 0, 3, 4, 4, 4, 4, 4, 4, ...
1, -4, 0, 4, 5, 5, 5, 5, 5, ...
1, -8, -9, 0, 5, 6, 6, 6, 6, ...
1, -8, -27, -14, 0, 6, 7, 7, 7, ...
1, 0, -54, -48, -20, 0, 7, 8, 8, ...
1, 16, -81, -116, -75, -27, 0, 8, 9, ...
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MATHEMATICA
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T[n_, k_] := Sum[(-1)^j * Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {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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