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A337085
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Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.
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4
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1, 0, 2, 0, 1, 5, 0, 1, 4, 16, 0, 1, 6, 15, 65, 0, 1, 10, 27, 64, 326, 0, 1, 18, 57, 124, 325, 1957, 0, 1, 34, 135, 292, 645, 1956, 13700, 0, 1, 66, 345, 796, 1585, 3906, 13699, 109601, 0, 1, 130, 927, 2404, 4605, 9726, 27391, 109600, 986410, 0, 1, 258, 2577, 7804, 15145, 28926, 68425, 219192, 986409, 9864101
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OFFSET
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0,3
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LINKS
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FORMULA
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T(0,k) = 0^k and T(n,k) = n^k + n * T(n-1,k) for n>0.
E.g.f. of column k: B_k(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024
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EXAMPLE
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Square array begins:
1, 0, 0, 0, 0, 0, 0, ...
2, 1, 1, 1, 1, 1, 1, ...
5, 4, 6, 10, 18, 34, 66, ...
16, 15, 27, 57, 135, 345, 927, ...
65, 64, 124, 292, 796, 2404, 7804, ...
326, 325, 645, 1585, 4605, 15145, 54645, ...
1957, 1956, 3906, 9726, 28926, 98646, 374526, ...
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MATHEMATICA
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T[n_, k_] := n! * Sum[If[j == k == 0, 1, j^k]/j!, {j, 0, n}]; Table[T[k, n-k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)
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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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