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A351703
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Square array T(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k * exp(x) / k!).
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6
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1, 1, 1, 1, 0, 4, 1, 0, 1, 21, 1, 0, 0, 3, 148, 1, 0, 0, 1, 12, 1305, 1, 0, 0, 0, 4, 70, 13806, 1, 0, 0, 0, 1, 10, 465, 170401, 1, 0, 0, 0, 0, 5, 40, 3591, 2403640, 1, 0, 0, 0, 0, 1, 15, 315, 31948, 38143377, 1, 0, 0, 0, 0, 0, 6, 35, 2296, 319068, 672552730
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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(0,k) = 1 and T(n,k) = binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * T(j,k) for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/k)} j^(n-k*j)/(k!^j * (n-k*j)!). - Seiichi Manyama, May 13 2022
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, ...
4, 1, 0, 0, 0, 0, ...
21, 3, 1, 0, 0, 0, ...
148, 12, 4, 1, 0, 0, ...
1305, 70, 10, 5, 1, 0, ...
13806, 465, 40, 15, 6, 1, ...
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PROG
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(PARI) T(n, k) = if(n==0, 1, binomial(n, k)*sum(j=0, n-k, binomial(n-k, j)*T(j, k)));
(PARI) T(n, k) = n!*sum(j=0, n\k, j^(n-k*j)/(k!^j*(n-k*j)!)); \\ Seiichi Manyama, May 13 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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