A291656 -id:A291656 - cp's OEIS frontend

A291676 Main diagonal of A291656.

0, 1, 10, 3527, 123296356, 757031629267449, 1263498691933197473321646, 823340843273442113630752833831086703, 285062591046216676379736060856308430344065653125000

Offset: 0

Seiichi Manyama, Aug 29 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..27

Cf. A291656, A335096.

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

a(n) = ((2n-1)!!)^n * Sum_{i=1..n} 1/(2*i-1)^n.

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

A335095 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.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Sep 12 2020

			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, ...

Column k=0..4 give A001477, A334670, A335090, A335091, A335092.
Rows n=0-2 give: A000004, A000012, A074606.
Main diagonal gives A335096.
Cf. A291656.

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 *)

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

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.

cp's OEIS Frontend

A291676 Main diagonal of A291656.

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