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A335975
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Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1) + x).
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4
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1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 15, 1, 1, 5, 19, 47, 52, 1, 1, 6, 29, 103, 227, 203, 1, 1, 7, 41, 189, 622, 1215, 877, 1, 1, 8, 55, 311, 1357, 4117, 7107, 4140, 1, 1, 9, 71, 475, 2576, 10589, 29521, 44959, 21147, 1, 1, 10, 89, 687, 4447, 23031, 88909, 227290, 305091, 115975, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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T(0,k) = 1 and T(n,k) = T(n-1,k) + k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.
T(n,k) = exp(-k) * Sum_{j>=0} (j + 1)^n * k^j / j!.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 5, 11, 19, 29, 41, 55, ...
1, 15, 47, 103, 189, 311, 475, ...
1, 52, 227, 622, 1357, 2576, 4447, ...
1, 203, 1215, 4117, 10589, 23031, 44683, ...
1, 877, 7107, 29521, 88909, 220341, 478207, ...
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MATHEMATICA
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T[0, k_] := 1; T[n_, k_] := T[n - 1, k] + k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)
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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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