A326855 -id:A326855 - cp's OEIS frontend

A326856 E.g.f.: Product_{k>=1} (1 + x^(4k-3) / (4k-3)).

1, 1, 0, 0, 0, 24, 144, 0, 0, 40320, 403200, 0, 0, 479001600, 8643317760, 29059430400, 0, 20922789888000, 475108274995200, 1871463083212800, 0, 2432902008176640000, 76354225980899328000, 525098781423304704000, 0, 620448401733239439360000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
Vaclav Kotesovec, Graph - the asymptotic ratio (40000 terms)

Cf. A007838, A088994, A326780, A326855.

nmax = 30; CoefficientList[Series[Product[(1+x^(4*k-3)/(4*k-3)), {k, 1, Floor[nmax/4]+1}], {x, 0, nmax}], x] * Range[0, nmax]!

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

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

A326856 E.g.f.: Product_{k>=1} (1 + x^(4k-3) / (4k-3)).

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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

A326856 E.g.f.: Product_{k>=1} (1 + x^(4*k-3) / (4*k-3)).

Views

Author

Keywords

Comments

Links

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)).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A326856 E.g.f.: Product_{k>=1} (1 + x^(4k-3) / (4k-3)).

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