A326755 -id:A326755 - cp's OEIS frontend

A326756 E.g.f.: Product_{k>=1} 1/(1 - x^(3k-2)/(3k-2)).

1, 1, 2, 6, 30, 150, 900, 7020, 58680, 528120, 5644080, 63510480, 769610160, 10483933680, 150733677600, 2272680828000, 37752297264000, 653710445308800, 11839468023187200, 231623795388268800, 4723930089495302400, 99779582243860358400, 2249431677071465356800

Offset: 0

Vaclav Kotesovec, Jul 23 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..447
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7).

Cf. A007841, A294506, A309319, A326755.

nmax = 25; CoefficientList[Series[1/Product[(1-x^(3*k-2)/(3*k-2)), {k, 1, Floor[nmax/3] + 1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ 3^(5/3) * exp(-gamma/3) * n^(1/3) * n! / Gamma(1/3)^2, where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function [Lehmer, 1972]. - Vaclav Kotesovec, Jul 23 2019

A326779 E.g.f.: Product_{k>=1} 1/(1 - x^(4k-1)/(4k-1)).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 80, 720, 0, 13440, 172800, 3628800, 5913600, 98841600, 4420915200, 92559667200, 110702592000, 6012444672000, 234205087334400, 6616915329024000, 13708373852160000, 771938716483584000, 40374130262409216000, 1172555787961958400000

Offset: 0

Vaclav Kotesovec, Jul 24 2019

nonn

In the article by Lehmer, Theorem 7, p. 387, case b <> 0 and b <> 1, correct formula is W_n(S_a,b) ~ a^(-1/a) * exp(-gamma/a) * (Gamma((b-1)/a) / (Gamma(b/a) * Gamma(1/a))) * n^(1/a - 1), where gamma is the Euler-Mascheroni constant (A001620) and Gamma() is the Gamma function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..449
Vaclav Kotesovec, Graph - the asymptotic ratio (50000 terms)
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7 needs a correction).

Cf. A007841, A294506, A309319, A326755, A326756, A326780.

nmax = 25; CoefficientList[Series[1/Product[(1-x^(4*k-1)/(4*k-1)), {k, 1, Floor[nmax/4] + 1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ exp(-gamma/4) * n! / (2 * sqrt(Pi) * n^(3/4)), where gamma is the Euler-Mascheroni constant A001620.

A326780 E.g.f.: Product_{k>=1} 1/(1 - x^(4k-3)/(4k-3)).

Original entry on oeis.org

1, 1, 2, 6, 24, 144, 864, 6048, 48384, 475776, 4902912, 53932032, 647184384, 8892398592, 126430875648, 1906924529664, 30510792474624, 539606261956608, 9890452422918144, 188459240926150656, 3773077461736095744, 81667528704634650624, 1819516013302975561728

Offset: 0

Vaclav Kotesovec, Jul 24 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..447
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7).

Cf. A007841, A294506, A309319, A326755, A326756, A326779.

nmax = 25; CoefficientList[Series[1/Product[(1-x^(4*k-3)/(4*k-3)), {k, 1, Floor[nmax/4] + 1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ 2^(7/2) * exp(-gamma/4) * n^(1/4) * n! / Gamma(1/4)^2, where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function [Lehmer, 1972].

A326857 E.g.f.: Product_{k>=1} (1 + x^(3k-1) / (3k-1)).

Original entry on oeis.org

1, 0, 1, 0, 0, 24, 0, 504, 5040, 0, 226800, 3628800, 0, 438721920, 6227020800, 16345929600, 1127656857600, 20922789888000, 58203397324800, 6697914906009600, 121645100408832000, 655224745383936000, 51359276952023040000, 1124000727777607680000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
Vaclav Kotesovec, Graph - the asymptotic ratio (30000 terms)

Cf. A007838, A088994, A326755, A326858.

nmax = 30; CoefficientList[Series[Product[(1+x^(3*k-1)/(3*k-1)), {k, 1, Floor[nmax/3]+1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ 2 * Pi * n! / (exp(gamma/3) * 3^(5/6) * Gamma(1/3)^2 * n^(2/3)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

A326860 E.g.f.: Product_{k>=1} (1 + x^(3k-1)/(3k-1)) / (1 - x^(3k-1)/(3k-1)).

Original entry on oeis.org

1, 0, 2, 0, 12, 48, 180, 2016, 15120, 72576, 1424304, 11249280, 113164128, 2066238720, 22751977248, 303261573888, 6400216892160, 85934653249536, 1440131337066240, 34330891188013056, 549029461368181248, 11212163885207900160, 296439802585781976576

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

In general, if c > 0, mod(d,c) <> 0, mod(d,c) <> 1 and e.g.f. = Product_{k>=1} (1 + x^(c*k+d)/(c*k+d)) / (1 - x^(c*k+d)/(c*k+d)), then a(n) ~ Gamma(1 + (d-1)/c) * n^(2/c - 1) * n! / (c^(2/c) * exp(2*gamma/c) * Gamma(2/c) * Gamma(1 + (d+1)/c)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

Cf. A305199, A326755, A326857, A326859, A326861.

nmax = 30; CoefficientList[Series[Product[(1+x^(3*k-1)/(3*k-1))/(1-x^(3*k-1)/(3*k-1)), {k, 1, Floor[nmax/3]+1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ exp(-2*gamma/3) * Gamma(1/3)^2 * n! / (2 * 3^(1/6) * Pi * n^(1/3)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

cp's OEIS Frontend

A326756 E.g.f.: Product_{k>=1} 1/(1 - x^(3*k-2)/(3*k-2)).

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A326779 E.g.f.: Product_{k>=1} 1/(1 - x^(4*k-1)/(4*k-1)).

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A326780 E.g.f.: Product_{k>=1} 1/(1 - x^(4*k-3)/(4*k-3)).

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A326857 E.g.f.: Product_{k>=1} (1 + x^(3*k-1) / (3*k-1)).

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A326860 E.g.f.: Product_{k>=1} (1 + x^(3*k-1)/(3*k-1)) / (1 - x^(3*k-1)/(3*k-1)).

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A326756 E.g.f.: Product_{k>=1} 1/(1 - x^(3k-2)/(3k-2)).

A326779 E.g.f.: Product_{k>=1} 1/(1 - x^(4k-1)/(4k-1)).

A326780 E.g.f.: Product_{k>=1} 1/(1 - x^(4k-3)/(4k-3)).

A326857 E.g.f.: Product_{k>=1} (1 + x^(3k-1) / (3k-1)).

A326860 E.g.f.: Product_{k>=1} (1 + x^(3k-1)/(3k-1)) / (1 - x^(3k-1)/(3k-1)).