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A292892
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Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (exp(x) - 1)).
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6
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1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 6, 16, 52, 1, 0, 0, 0, 12, 65, 203, 1, 0, 0, 0, 24, 20, 336, 877, 1, 0, 0, 0, 0, 60, 390, 1897, 4140, 1, 0, 0, 0, 0, 120, 120, 2562, 11824, 21147, 1, 0, 0, 0, 0, 0, 360, 210, 11816, 80145, 115975, 1, 0, 0, 0, 0, 0, 720, 840, 20496, 105912, 586000, 678570
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OFFSET
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0,6
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LINKS
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FORMULA
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T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(n-k*j)!.
T(0,k) = 1 and T(n,k) = (n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, ...
2, 2, 0, 0, 0, ...
5, 3, 6, 0, 0, ...
15, 16, 12, 24, 0, ...
52, 65, 20, 60, 120, ...
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PROG
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(PARI) T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(n-k*j)!); \\ Seiichi Manyama, Jul 09 2022
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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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