A285289 -id:A285289 - cp's OEIS frontend

A263345 Expansion of Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^k.

1, 1, 2, 4, 7, 14, 22, 40, 65, 107, 176, 282, 448, 705, 1101, 1701, 2611, 3977, 6021, 9048, 13527, 20102, 29720, 43712, 63997, 93259, 135317, 195539, 281440, 403559, 576568, 820888, 1164826, 1647583, 2323169, 3266041, 4578305, 6399990, 8922389, 12406535

Offset: 0

Vaclav Kotesovec, Oct 15 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A003105, A026007, A262876, A262877, A262878, A262879, A263346.

Cf. A285289, A285290, A285291.

nmax=40; CoefficientList[Series[Product[((1 + x^k)/(1 + x^(3*k)))^k,{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(1/6) * exp(2^(-1/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(1/6) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

A285290 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(4*k)))^k.

Original entry on oeis.org

1, 1, 2, 5, 7, 15, 26, 44, 74, 125, 205, 331, 534, 844, 1332, 2077, 3215, 4934, 7533, 11410, 17191, 25751, 38346, 56833, 83814, 123025, 179776, 261639, 379186, 547476, 787516, 1128775, 1612395, 2295701, 3258177, 4610130, 6503873, 9149365, 12835612, 17959085

Offset: 0

Vaclav Kotesovec, Apr 16 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A285289, A263345, A285291.

nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1+x^(4*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2^(-8/3) * 3^(5/3) * (5*Zeta(3))^(1/3) * n^(2/3)) * (5*Zeta(3))^(1/6) / (2^(4/3) * 3^(1/6) * sqrt(Pi) * n^(2/3)).

A285291 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(5*k)))^k.

Original entry on oeis.org

1, 1, 2, 5, 8, 15, 27, 47, 78, 134, 218, 356, 576, 916, 1449, 2268, 3525, 5431, 8324, 12652, 19129, 28754, 42974, 63898, 94553, 139241, 204144, 298045, 433328, 627592, 905560, 1301934, 1865362, 2663816, 3791813, 5380911, 7613286, 10740839, 15111141, 21202615

Offset: 0

Vaclav Kotesovec, Apr 16 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} ((1 + x^k) / (1 + x^(m*k)))^k, then a(n, m) ~ exp(2^(-4/3) * 3^(4/3) * (1-1/m^2)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * ((1-1/m^2)*Zeta(3))^(1/6) / (2^(2/3) * 3^(1/3) * sqrt(Pi) * n^(2/3)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A285289 (m=2), A263345 (m=3), A285290 (m=4).

nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1+x^(5*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2^(-1/3) * 3^(5/3) * 5^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (5^(1/3) * 6^(1/6) * sqrt(Pi) * n^(2/3)).

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A263345 Expansion of Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^k.

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A285290 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(4*k)))^k.

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A285291 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(5*k)))^k.

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