A326860 -id:A326860 - cp's OEIS frontend

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

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.

A326862 E.g.f.: Product_{k>=1} (1 + x^(4k-1)/(4k-1)) / (1 - x^(4k-1)/(4k-1)).

Original entry on oeis.org

1, 0, 0, 4, 0, 0, 160, 1440, 0, 26880, 691200, 7257600, 11827200, 395366400, 14125363200, 185119334400, 442810368000, 24049778688000, 919255538073600, 13662913904640000, 54833495408640000, 3627817738960896000, 142917996623560704000, 2442221696292618240000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A326855, A326779.

Cf. A305199, A326859, A326860, A326863.

nmax = 30; CoefficientList[Series[Product[(1+x^(4*k-1)/(4*k-1))/(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/2) * n! / (2*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

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

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A326862 E.g.f.: Product_{k>=1} (1 + x^(4k-1)/(4k-1)) / (1 - x^(4k-1)/(4k-1)).

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A326862 E.g.f.: Product_{k>=1} (1 + x^(4*k-1)/(4*k-1)) / (1 - x^(4*k-1)/(4*k-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

A326862 E.g.f.: Product_{k>=1} (1 + x^(4k-1)/(4k-1)) / (1 - x^(4k-1)/(4k-1)).