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A343863
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (n!)^k * Sum_{j=1..n} (1/j!)^k.
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2
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1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 9, 16, 5, 1, 2, 17, 82, 65, 6, 1, 2, 33, 460, 1313, 326, 7, 1, 2, 65, 2674, 29441, 32826, 1957, 8, 1, 2, 129, 15796, 684545, 3680126, 1181737, 13700, 9, 1, 2, 257, 94042, 16175105, 427840626, 794907217, 57905114, 109601, 10
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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) = 1 and T(n,k) = n^k * T(n-1,k) + 1 for n > 0.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, ...
3, 5, 9, 17, 33, 65, ...
4, 16, 82, 460, 2674, 15796, ...
5, 65, 1313, 29441, 684545, 16175105, ...
6, 326, 32826, 3680126, 427840626, 50547203126, ...
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MATHEMATICA
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T[n_, k_] := Sum[(n!/j!)^k, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 03 2021 *)
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PROG
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(PARI) T(n, k) = sum(j=0, n, (n!/j!)^k);
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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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