A304625 -id:A304625 - cp's OEIS frontend

A270915 Decimal expansion of a constant related to the asymptotics of A008485.

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

Offset: 1

Vaclav Kotesovec, Mar 25 2016

			5.352701333486642687772415814165327879851483271286947097319690756...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A008485, A109084, A109085, A206228, A270914, A303070, A304625, A327279.

RealDigits[1/r /. FindRoot[{s == 1/QPochhammer[r*s], QPochhammer[r*s] + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Equals limit n->infinity A008485(n)^(1/n).

A304626 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(n*k)))^n.

Original entry on oeis.org

1, 0, 1, 10, 47, 201, 849, 3578, 15147, 64516, 276268, 1188342, 5130987, 22226036, 96543989, 420368843, 1834203939, 8018057328, 35107961157, 153950675566, 675978772306, 2971700764920, 13078268135661, 57613905606250, 254038914924767, 1121081799217206, 4951199308679965

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Number of partitions of n into 2 or more distinct parts, with n types of each part. - Ilya Gutkovskiy, May 16 2018

Index entries for sequences related to partitions

Cf. A058576, A073252, A111133, A255526, A270913, A304625.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(n k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[(1 + x^k)^n, {k, 1, n - 1}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[(QPochhammer[-1, x, 1 + n]/QPochhammer[-1, x^n, 1 + n])^n, {x, 0, n}], {n, 0, 26}]

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.502476747617354487738... and c = 0.2605422331424384694... - Vaclav Kotesovec, May 16 2018

A319670 a(n) = [x^n] Product_{k>=2} 1/(1 - x^k)^n.

Original entry on oeis.org

1, 0, 2, 3, 14, 30, 119, 301, 1078, 3036, 10242, 30624, 100451, 310128, 1004817, 3158343, 10182982, 32345186, 104145896, 332953929, 1072383374, 3442913407, 11100120528, 35742258497, 115377720235, 372326184555, 1203406838428, 3890040945078, 12588182588373, 40748118469180

Offset: 0

Ilya Gutkovskiy, Sep 25 2018

nonn

Number of partitions of n into parts > 1, if there are n kinds of parts.

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

Cf. A000203, A002865, A008485, A147766, A191659, A304625, A319671.

Table[SeriesCoefficient[Product[1/(1 - x^k)^n , {k, 2, n}], {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[((1 - x)/QPochhammer[x])^n, {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, k] - 1) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]

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

a(n) ~ c * d^n / sqrt(n), where d = 3.293558598422332665054219310876308... and c = 0.2154241499279313950113565475... - Vaclav Kotesovec, Oct 06 2018

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A270915 Decimal expansion of a constant related to the asymptotics of A008485.

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A304626 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(n*k)))^n.

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A319670 a(n) = [x^n] Product_{k>=2} 1/(1 - x^k)^n.

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