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A335095
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Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n+1)!!)^k * Sum_{j=1..n} 1/(2*j+1)^k.
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5
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0, 0, 1, 0, 1, 2, 0, 1, 8, 3, 0, 1, 34, 71, 4, 0, 1, 152, 1891, 744, 5, 0, 1, 706, 55511, 164196, 9129, 6, 0, 1, 3368, 1745731, 41625144, 20760741, 129072, 7, 0, 1, 16354, 57365351, 11575291716, 56246975289, 3616621254, 2071215, 8
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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) = 0, T(1,k) = 1 and T(n,k) = ((2*n-1)^k+(2*n+1)^k) * T(n-1,k) - (2*n-1)^(2*k) * T(n-2, k) for n>1.
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EXAMPLE
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Square array begins:
0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, ...
2, 8, 34, 152, 706, ...
3, 71, 1891, 55511, 1745731, ...
4, 744, 164196, 41625144, 11575291716, ...
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MATHEMATICA
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T[n_, k_] := ((2*n + 1)!!)^k * Sum[1/(2*j + 1)^k, {j, 1, n}]; Table[T[k, n - k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)
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PROG
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(PARI) {T(n, k) = prod(j=1, n, 2*j+1)^k*sum(j=1, n, 1/(2*j+1)^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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