A303071 -id:A303071 - cp's OEIS frontend

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

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

Offset: 1

Vaclav Kotesovec, Mar 25 2016

This constant is very close to exp(5*Pi/(6*sqrt(2))) / sqrt(2) = 4.502476748630924546525119125234175537729... - Vaclav Kotesovec, May 17 2018

			4.502476747617354487738593932700784406763128756091621633464540424...

Cf. A181315, A206229, A270913, A270915, A303071, A304626, A327280.

RealDigits[1/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, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Equals limit n->infinity A270913(n)^(1/n).

A304782 a(n) = [x^n] (1/(1 - x))Product_{k>=1} (1 + nx^k).

Original entry on oeis.org

1, 2, 5, 19, 49, 126, 469, 1177, 2881, 6481, 23101, 53725, 127153, 274288, 581925, 1860751, 4155649, 9279791, 19409221, 39839239, 77052401, 229393207, 481747949, 1035561408, 2082441025, 4153434376, 7822058869, 14686515649, 39394280689, 79657493191, 163600884901

Offset: 0

Ilya Gutkovskiy, May 18 2018

nonn

Index entries for sequences related to partitions

Cf. A286957, A291698, A303070, A303071, A303914.

Table[SeriesCoefficient[1/(1 - x) Product[(1 + n x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/(1 - x) Exp[Sum[(-1)^(k + 1) n^k x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[QPochhammer[-n, x]/((1 + n) (1 - x)), {x, 0, n}], {n, 0, 30}]

a(n) = [x^n] (1/(1 - x))*exp(Sum_{k>=1} (-1)^(k+1)*n^k*x^k/(k*(1 - x^k))).

a(n) = Sum_{j=0..n} A286957(j,n).

cp's OEIS Frontend

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

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A304782 a(n) = [x^n] (1/(1 - x))Product_{k>=1} (1 + nx^k).

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cp's OEIS Frontend

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A304782 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + n*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A304782 a(n) = [x^n] (1/(1 - x))Product_{k>=1} (1 + nx^k).