A326857 -id:A326857 - cp's OEIS frontend

A326855 E.g.f.: Product_{k>=1} (1 + x^(4k-1) / (4k-1)).

1, 0, 0, 2, 0, 0, 0, 720, 0, 0, 172800, 3628800, 0, 0, 2641766400, 87178291200, 0, 0, 225422681702400, 6402373705728000, 0, 221172909834240000, 30424079849619456000, 1124000727777607680000, 0, 49241936645495193600000, 11321261082950211993600000

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 (40000 terms)

Cf. A007838, A088994, A326779, A326856, A326857.

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

a(n) ~ Pi * n! / (exp(gamma/4) * Gamma(1/4)^2 * n^(3/4)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

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.

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

A326855 E.g.f.: Product_{k>=1} (1 + x^(4k-1) / (4k-1)).

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

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

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cp's OEIS Frontend

A326855 E.g.f.: Product_{k>=1} (1 + x^(4*k-1) / (4*k-1)).

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

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Formula

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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A326855 E.g.f.: Product_{k>=1} (1 + x^(4k-1) / (4k-1)).

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

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