A327214 -id:A327214 - cp's OEIS frontend

A327280 Decimal expansion of a constant related to A270913 and A327214.

2, 6, 0, 5, 4, 2, 2, 3, 3, 1, 4, 2, 4, 3, 8, 4, 6, 9, 4, 3, 3, 8, 6, 0, 8, 3, 2, 1, 6, 0, 4, 2, 2, 4, 8, 7, 6, 7, 6, 9, 0, 2, 0, 6, 0, 3, 9, 0, 8, 1, 6, 3, 0, 9, 2, 6, 3, 0, 2, 5, 9, 0, 1, 0, 2, 4, 1, 0, 3, 7, 5, 7, 2, 8, 9, 6, 2, 5, 4, 2, 2, 7, 8, 3, 5, 6, 5, 0, 8, 6, 0, 4, 0, 4, 4, 3, 1, 1, 4, 1, 0, 1, 0, 6, 2, 6

Offset: 0

Vaclav Kotesovec, Aug 28 2019

			0.26054223314243846943386083216042248767690206039081630926302590102410375728962...

Cf. A270913, A270914, A327214, A327279, A366018.

RealDigits[1/Sqrt[s*Pi*Derivative[0, 2][QPochhammer][-1, r*s]]/r /. FindRoot[{2*s == QPochhammer[-1, r*s], r*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/2}, {s, 1/2}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 02 2023 *)

Equals limit_{n->infinity} A270913(n) * sqrt(n) / A270914^n.

A309986 Convolution of A270913 and A008485.

Original entry on oeis.org

1, 2, 9, 43, 206, 999, 4915, 24372, 121698, 611244, 3085612, 15645347, 79639602, 406809249, 2084567381, 10712007629, 55187254157, 284981396231, 1474729519719, 7646180479889, 39713643612380, 206600871071930, 1076372569004514, 5615363541987786, 29331204404385053

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A008485, A270913, A270919, A327214, A327215.

A270913[n_] := SeriesCoefficient[Product[(1 + x^k)^n, {k, 1, n}], {x, 0, n}];
A008485[n_] := SeriesCoefficient[Product[1/(1 - x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A008485[k]*A270913[n-k], {k, 0, n}], {n, 0, 20}]

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687... and c = 0.446705640528056029457240607298917821281915554...

A327215 Self-convolution of A008485.

Original entry on oeis.org

1, 2, 11, 54, 279, 1442, 7530, 39474, 207693, 1095522, 5790116, 30650038, 162451560, 861920492, 4577055823, 24323292984, 129338944225, 688128700422, 3662798123481, 19504467792378, 103899170100154, 553642311668244, 2951010332435759, 15733439067954134

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000712, A008485, A304444, A309986, A327214.

A008485[n_]:=SeriesCoefficient[Product[1/(1-x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A008485[k]*A008485[n-k], {k, 0, n}], {n, 0, 25}]

a(n) ~ c^2 * Pi * d^n, where d = A270915 = 5.352701333486642687772415814165... and c = A327279 = 0.26801521271073331568695383828... (see A008485).

cp's OEIS Frontend

A327280 Decimal expansion of a constant related to A270913 and A327214.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A309986 Convolution of A270913 and A008485.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A327215 Self-convolution of A008485.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula