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A330843
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Square array T(n,k) = [x^n] ((1+x)^(k+1) / (1-x)^(k-1))^n, n>=0, k>=0, read by descending antidiagonals.
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4
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1, 1, 0, 1, 2, -2, 1, 4, 6, 0, 1, 6, 30, 20, 6, 1, 8, 70, 256, 70, 0, 1, 10, 126, 924, 2310, 252, -20, 1, 12, 198, 2240, 12870, 21504, 924, 0, 1, 14, 286, 4420, 41990, 184756, 204204, 3432, 70, 1, 16, 390, 7680, 104006, 811008, 2704156, 1966080, 12870, 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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T(n,k) = Sum_{j=0..n} binomial((k+1)*n,j) * binomial(k*n-j-1,n-j).
T(n,k) = 1/n! * ((k+1)*n)!/Gamma(1 + (k+1)*n/2) * Gamma(1 + (k-1)*n/2)/((k-1)*n)!.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, ...
-2, 6, 30, 70, 126, 198, ...
0, 20, 256, 924, 2240, 4420, ...
6, 70, 2310, 12870, 41990, 104006, ...
0, 252, 21504, 184756, 811008, 2521260, ...
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MATHEMATICA
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T[n_, k_] := Sum[Binomial[(k + 1)*n, j] * Binomial[k*n - j - 1, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 05 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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