A291585 -id:A291585 - 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).

A335091 a(n) = ((2n+1)!!)^3 (Sum_{k=1..n} 1/(2*k+1)^3).

Original entry on oeis.org

0, 1, 152, 55511, 41625144, 56246975289, 124697847089808, 423322997436687375, 2088114588247920714000, 14363296872939657999716625, 133299155158711610547152961000, 1624450039177408057102079622846375, 25413656551949715361011431877529125000, 500711137690193661025654228810320074015625

Offset: 0

Seiichi Manyama, Sep 11 2020

nonn

Column k=3 of A335095.

Cf. A271636, A291585, A334670, A335090, A335092.

a[n_] := ((2*n + 1)!!)^3 * Sum[1/(2*k + 1)^3, {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Apr 29 2021 *)

{a(n) = prod(k=1, n, 2*k+1)^3*sum(k=1, n, 1/(2*k+1)^3)}

{a(n) = if(n<2, n, ((2*n-1)^3+(2*n+1)^3)*a(n-1)-(2*n-1)^6*a(n-2))}

a(n) = ((2*n-1)^3+(2*n+1)^3) * a(n-1) - (2*n-1)^6 * a(n-2) for n>1.

a(n) ~ (7*zeta(3)/8 - 1) * 2^(3*n + 9/2) * n^(3*n + 3) / exp(3*n). - Vaclav Kotesovec, Sep 25 2020

A291587 a(n) = ((2n-1)!!)^5 * Sum_{i=1..n} 1/(2*i-1)^5.

Original entry on oeis.org

0, 1, 244, 762743, 12820180976, 757031629267449, 121921454556651769524, 45268703999809586294371407, 34375967164840303438628549400000, 48808991831991566280900452880679940625, 120855944455445379138034328603009420077012500

Offset: 0

Seiichi Manyama, Aug 27 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..105

Cf. A001147, A099827.

Cf. A004041, A001824, A291585, A291586.

Table[(2*n-1)!!^5 * Sum[1/(2*i-1)^5, {i, 1, n}], {n, 0, 12}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(0) = 0, a(1) = 1, a(n+1) = ((2*n-1)^5+(2*n+1)^5)*a(n) - (2*n-1)^10*a(n-1) for n > 0.

a(n) ~ 31*Zeta(5) * 2^(5*n-5/2) * n^(5*n) / exp(5*n). - Vaclav Kotesovec, Aug 27 2017

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

A335091 a(n) = ((2n+1)!!)^3 (Sum_{k=1..n} 1/(2*k+1)^3).

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Author

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Mathematica

PARI

PARI

Formula

A291587 a(n) = ((2n-1)!!)^5 * Sum_{i=1..n} 1/(2*i-1)^5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

A335091 a(n) = ((2*n+1)!!)^3 * (Sum_{k=1..n} 1/(2*k+1)^3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A291587 a(n) = ((2n-1)!!)^5 * Sum_{i=1..n} 1/(2*i-1)^5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A335091 a(n) = ((2n+1)!!)^3 (Sum_{k=1..n} 1/(2*k+1)^3).