A326756 -id:A326756 - cp's OEIS frontend

A326755 E.g.f.: Product_{k>=1} 1/(1 - x^(3k-1)/(3k-1)).

1, 0, 1, 0, 6, 24, 90, 504, 7560, 18144, 485352, 4626720, 32033232, 516559680, 9142044912, 64700161344, 1804378343040, 29722011830784, 308081755013760, 8202581858225664, 184073277074529024, 2067986628774743040, 75069447974837132544, 1673053361596502645760

Offset: 0

Vaclav Kotesovec, Jul 23 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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..448
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7 needs a correction).

Cf. A007841, A294506, A309319, A326756.

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

a(n) ~ 3^(1/6) * exp(-gamma/3) * Gamma(1/3) * n! / (2*Pi*n^(2/3)).

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

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].

A326858 E.g.f.: Product_{k>=1} (1 + x^(3k-2) / (3k-2)).

Original entry on oeis.org

1, 1, 0, 0, 6, 30, 0, 720, 5760, 0, 362880, 5417280, 17107200, 479001600, 8885479680, 32691859200, 1307674368000, 34151856076800, 214585052774400, 6402373705728000, 192796754895360000, 1542202547010048000, 55105230485200896000, 1944933030182596608000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A007838, A088994, A326756, A326857.

nmax = 30; CoefficientList[Series[Product[(1+x^(3*k-2)/(3*k-2)), {k, 1, Floor[nmax/3]+1}], {x, 0, nmax}], x] * Range[0, nmax]!

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

A326861 E.g.f.: Product_{k>=1} (1 + x^(3k-2)/(3k-2)) / (1 - x^(3k-2)/(3k-2)).

Original entry on oeis.org

1, 2, 4, 12, 60, 360, 2160, 16560, 149040, 1386720, 14592960, 174208320, 2173897440, 29413264320, 437473872000, 6792952636800, 112213292716800, 2002551280012800, 37194983281843200, 726119227314201600, 15112608758893324800, 326665495054151193600

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

In general, if c > 0, d = 1-c 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) ~ 2 * n^(2/c) * n! / (c^(2/c) * exp(2*gamma/c) * Gamma(1 + 2/c)^2), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

Cf. A305199, A326756, A326858, A326860.

nmax = 30; CoefficientList[Series[Product[(1+x^(3*k-2)/(3*k-2))/(1-x^(3*k-2)/(3*k-2)), {k, 1, Floor[nmax/3]+1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ 3^(7/3) * exp(-2*gamma/3) * Gamma(1/3)^2 * n^(2/3) * n! / (8 * Pi^2), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

cp's OEIS Frontend

A326755 E.g.f.: Product_{k>=1} 1/(1 - x^(3*k-1)/(3*k-1)).

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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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A326858 E.g.f.: Product_{k>=1} (1 + x^(3*k-2) / (3*k-2)).

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A326861 E.g.f.: Product_{k>=1} (1 + x^(3*k-2)/(3*k-2)) / (1 - x^(3*k-2)/(3*k-2)).

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A326755 E.g.f.: Product_{k>=1} 1/(1 - x^(3k-1)/(3k-1)).

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)).

A326858 E.g.f.: Product_{k>=1} (1 + x^(3k-2) / (3k-2)).

A326861 E.g.f.: Product_{k>=1} (1 + x^(3k-2)/(3k-2)) / (1 - x^(3k-2)/(3k-2)).