A263345 -id:A263345 - cp's OEIS frontend

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

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)).

A263346 Expansion of Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^k.

Original entry on oeis.org

1, 1, 3, 5, 12, 21, 40, 71, 130, 221, 387, 648, 1095, 1800, 2964, 4792, 7730, 12301, 19510, 30619, 47859, 74179, 114469, 175427, 267684, 406039, 613325, 921671, 1379500, 2055313, 3050652, 4509385, 6641966, 9746452, 14254242, 20775255, 30184451, 43715711

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. A000726, A000219, A262876, A262877, A262878, A262879, A263345.

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

a(n) ~ 2^(1/6) * Zeta(3)^(1/6) * exp(6^(1/3) * Zeta(3)^(1/3) * n^(2/3)) / (3^(11/12) * sqrt(Pi) * n^(2/3)).

A285289 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(2*k)))^k.

Original entry on oeis.org

1, 1, 1, 4, 5, 10, 16, 26, 44, 68, 110, 167, 265, 399, 609, 919, 1371, 2040, 3005, 4420, 6436, 9364, 13501, 19433, 27806, 39639, 56265, 79572, 112126, 157390, 220283, 307163, 427145, 592029, 818359, 1127878, 1550483, 2125656, 2907013, 3965853, 5397497

Offset: 0

Vaclav Kotesovec, Apr 16 2017

nonn

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

Cf. A263345, A285290, A285291.

nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1+x^(2*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

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

cp's OEIS Frontend

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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A263346 Expansion of Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^k.

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A285289 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(2*k)))^k.

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