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A308484
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Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. log(1 + Sum_{j>=1} j^k * x^j/j!).
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1
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1, 1, 0, 1, 1, 0, 1, 3, -1, 0, 1, 7, -1, -2, 0, 1, 15, 5, -26, 9, 0, 1, 31, 35, -146, 29, 6, 0, 1, 63, 149, -650, -351, 756, -155, 0, 1, 127, 539, -2642, -5251, 9936, -1793, 232, 0, 1, 255, 1805, -10346, -46071, 83376, 51421, -45744, 3969, 0
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OFFSET
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1,8
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LINKS
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FORMULA
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A(n,k) = n^k - Sum_{j=1..n-1} binomial(n-1,j)*j^k*A(n-j,k).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 3, 7, 15, 31, ...
0, -1, -1, 5, 35, 149, ...
0, -2, -26, -146, -650, -2642, ...
0, 9, 29, -351, -5251, -46071, ...
0, 6, 756, 9936, 83376, 559656, ...
0, -155, -1793, 51421, 1623439, 28735405, ...
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MATHEMATICA
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T[n_, k_] := T[n, k] = n^k - Sum[Binomial[n-1, j] * j^k * T[n-j, k], {j, 1, n-1}]; Table[T[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 12 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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