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A295181
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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)/(1 - x)^k.
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1
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1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 3, 4, 9, 0, 1, 0, 4, 6, 24, 44, 0, 1, 0, 5, 8, 45, 128, 265, 0, 1, 0, 6, 10, 72, 252, 880, 1854, 0, 1, 0, 7, 12, 105, 416, 1935, 6816, 14833, 0, 1, 0, 8, 14, 144, 620, 3520, 16146, 60032, 133496, 0, 1, 0, 9, 16, 189, 864, 5725, 31104, 153657, 589312, 1334961, 0
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OFFSET
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0,13
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COMMENTS
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A(n,k) is the k-fold exponential convolution of A000166 with themselves, 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)/(1 - x)^k.
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EXAMPLE
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E.g.f. of column k: A_k(x) = 1 + k*x^2/2! + 2*k*x^3/3! + 3*k*(k + 2)*x^4/4! + 4*k*(5*k + 6)*x^5/5! + 5*k*(3*k^2 + 26*k + 24)*x^6/6! + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 0, 0, 0, 0, 0, ...
0, 1, 2, 3, 4, 5, ...
0, 2, 4, 6, 8, 10, ...
0, 9, 24, 45, 72, 105, ...
0, 44, 128, 252, 416, 620, ...
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MATHEMATICA
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Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
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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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