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A357869
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j)/j!.
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3
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1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 5, 0, 1, 0, 0, 6, 15, 0, 1, 0, 0, 6, 26, 52, 0, 1, 0, 0, 0, 36, 150, 203, 0, 1, 0, 0, 0, 24, 150, 962, 877, 0, 1, 0, 0, 0, 0, 240, 900, 6846, 4140, 0, 1, 0, 0, 0, 0, 120, 1560, 9366, 54266, 21147, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 101556, 471750, 115975, 0
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OFFSET
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0,9
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LINKS
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FORMULA
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For k > 0, e.g.f. of column k: exp((exp(x) - 1)^k).
T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * Stirling2(j,k) * T(n-j,k).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 0, 0, 0, 0, ...
0, 2, 2, 0, 0, 0, ...
0, 5, 6, 6, 0, 0, ...
0, 15, 26, 36, 24, 0, ...
0, 52, 150, 150, 240, 120, ...
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PROG
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(PARI) T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2)/j!);
(PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(exp((exp(x+x*O(x^n))-1)^k), n));
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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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