A326858 -id:A326858 - cp's OEIS frontend

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

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.

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

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

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

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cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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

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