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A308497
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Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. log(1 + Sum_{j>=1} binomial(j+k-1,k) * x^j/j).
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4
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1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 8, 1, 4, 10, 15, 24, 26, 1, 5, 17, 34, 54, 120, 194, 1, 6, 26, 69, 104, 240, 720, 1142, 1, 7, 37, 126, 204, 200, 1350, 5040, 9736, 1, 8, 50, 211, 408, -330, -400, 9450, 40320, 81384, 1, 9, 65, 330, 794, -1704, -12510, -2800, 78120, 362880, 823392
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OFFSET
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1,8
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COMMENTS
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Column k > 2 is asymptotic to -2*(n-1)! * cos(n*arctan(sin(Pi/k)/(cos(Pi/k) - (k-1)^(1/k)))) / (1 + 1/(k-1)^(2/k) - 2*cos(Pi/k)/(k-1)^(1/k))^(n/2). - Vaclav Kotesovec, May 12 2021
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LINKS
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FORMULA
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A(n,k) = (1/k!) * ((n+k-1)! - Sum_{j=1..n-1} binomial(n-1,j) * (j+k-1)! * A(n-j,k)).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
1, 2, 5, 10, 17, 26, ...
1, 6, 15, 34, 69, 126, ...
8, 24, 54, 104, 204, 408, ...
26, 120, 240, 200, -330, -1704, ...
194, 720, 1350, -400, -12510, -51696, ...
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MATHEMATICA
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T[n_, k_] := T[n, k] = ((n+k-1)! - Sum[Binomial[n-1, j] * (j+k-1)! * T[n-j, k], {j, 1, n-1}])/k!; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 12 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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