A291676 -id:A291676 - cp's OEIS frontend

A291656 Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n-1)!!)^k * Sum_{i=1..n} 1/(2*i-1)^k.

0, 0, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 10, 23, 4, 0, 1, 28, 259, 176, 5, 0, 1, 82, 3527, 12916, 1689, 6, 0, 1, 244, 51331, 1213136, 1057221, 19524, 7, 0, 1, 730, 762743, 123296356, 885533769, 128816766, 264207, 8, 0, 1, 2188, 11406979, 12820180976, 809068942341, 1179489355164, 21878089479, 4098240, 9

Offset: 0

Seiichi Manyama, Aug 28 2017

			Square array begins:
  0,   0,     0,       0,         0, ...
  1,   1,     1,       1,         1, ...
  2,   4,    10,      28,        82, ...
  3,  23,   259,    3527,     51331, ...
  4, 176, 12916, 1213136, 123296356, ...

Seiichi Manyama, Antidiagonals n = 0..54, flattened

Columns k=0-5 give: A001477, A004041(n+1), A001824(n+1), A291585, A291586, A291587.
Rows n=0-2 give: A000004, A000012, A034472.
Main diagonal gives A291676.
Cf. A291556.

T(0,k) = 0, T(1,k) = 1 and T(n+1, k) = ((2*n-1)^k+(2*n+1)^k) * T(n, k) - (2*n-1)^(2*k) * T(n-1, k).

A335096 a(n) = ((2n+1)!!)^n (Sum_{k=1..n} 1/(2*k+1)^n).

Original entry on oeis.org

0, 1, 34, 55511, 11575291716, 548347875819272649, 8811385079228718926321932614, 66303398534111438173105653188803948359375, 308529654991526005900670429792887300937160115962403125000

Offset: 0

Seiichi Manyama, Sep 12 2020

nonn

Main diagonal of A335095.

Cf. A291676.

Table[((2*n + 1)!!)^n * Sum[1/(2*k + 1)^n, {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Sep 25 2020 *)

{a(n) = prod(k=1, n, 2*k+1)^n*sum(k=1, n, 1/(2*k+1)^n)}

a(n) ~ 2^(n*(n + 3/2)) * n^(n*(n+1)) / (3^n * exp(n^2 - 11/24)). - Vaclav Kotesovec, Sep 25 2020

cp's OEIS Frontend

A291656 Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n-1)!!)^k * Sum_{i=1..n} 1/(2*i-1)^k.

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Formula

A335096 a(n) = ((2n+1)!!)^n (Sum_{k=1..n} 1/(2*k+1)^n).

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Author

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Mathematica

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Formula

cp's OEIS Frontend

A291656 Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n-1)!!)^k * Sum_{i=1..n} 1/(2*i-1)^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A335096 a(n) = ((2*n+1)!!)^n * (Sum_{k=1..n} 1/(2*k+1)^n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A335096 a(n) = ((2n+1)!!)^n (Sum_{k=1..n} 1/(2*k+1)^n).