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A292915
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Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(2 - exp(x)).
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1
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1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 11, 26, 75, 1, 5, 18, 51, 150, 541, 1, 6, 27, 94, 299, 1082, 4683, 1, 7, 38, 161, 582, 2163, 9366, 47293, 1, 8, 51, 258, 1083, 4294, 18731, 94586, 545835, 1, 9, 66, 391, 1910, 8345, 37398, 189171, 1091670, 7087261, 1, 10, 83, 566, 3195, 15666, 74067, 378214, 2183339, 14174522, 102247563
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OFFSET
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0,5
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COMMENTS
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A(n,k) is the k-th binomial transform of A000670 evaluated at n.
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LINKS
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FORMULA
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E.g.f. of column k: exp(k*x)/(2 - exp(x)).
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EXAMPLE
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E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 3)*x^2/2! + (k^3 + 3*k^2 + 9*k + 13)*x^3/3! + (k^4 + 4*k^3 + 18*k^2 + 52*k + 75) x^4/4! + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
3, 6, 11, 18, 27, 38, ...
13, 26, 51, 94, 161, 258, ...
75, 150, 299, 582, 1083, 1910, ...
541, 1082, 2163, 4294, 8345, 15666, ...
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MAPLE
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A:= proc(n, k) option remember; k^n +add(
binomial(n, j)*A(j, k), j=0..n-1)
end:
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MATHEMATICA
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Table[Function[k, n! SeriesCoefficient[Exp[k x]/(2 - Exp[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, HurwitzLerchPhi[1/2, -n, k]/2][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
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PROG
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(PARI) a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
A(n, k) = 2^k*a000670(n)-sum(j=0, k-1, 2^j*(k-1-j)^n); \\ Seiichi Manyama, Dec 25 2023
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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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