A263346 -id:A263346 - 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)).

A285262 Expansion of Product_{k>=1} ((1-x^(4*k))/(1-x^k))^k.

Original entry on oeis.org

1, 1, 3, 6, 12, 23, 45, 80, 145, 256, 446, 761, 1292, 2154, 3568, 5842, 9485, 15261, 24386, 38647, 60867, 95212, 148052, 228860, 351899, 538186, 819105, 1240704, 1870886, 2808888, 4199880, 6254577, 9279179, 13715740, 20202040, 29654210, 43386131, 63274874

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

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

Cf. A026007, A263346, A285263.

Cf. A285215.

nmax = 40; CoefficientList[Series[Product[((1-x^(4*k))/(1-x^k))^k, {k,1,nmax}], {x,0,nmax}], x]

a(n) ~ exp(3^(4/3) * (5*Zeta(3))^(1/3) * n^(2/3) / 4) * (5*Zeta(3))^(1/6) / (2^(7/6) * 3^(1/3)* sqrt(Pi) * n^(2/3)).

A285263 Expansion of Product_{k>=1} ((1-x^(5*k))/(1-x^k))^k.

Original entry on oeis.org

1, 1, 3, 6, 13, 23, 47, 83, 154, 269, 474, 809, 1387, 2313, 3859, 6330, 10341, 16680, 26790, 42586, 67375, 105731, 165097, 256052, 395248, 606501, 926502, 1408048, 2130788, 3209643, 4815595, 7194875, 10709843, 15881236, 23467805, 34556842, 50720003, 74200845

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} ((1-x^(m*k))/(1-x^k))^k, then a(n, m) ~ exp(3 * 2^(-2/3) * ((1-1/m^2)*Zeta(3))^(1/3) * n^(2/3)) * ((1-1/m^2)*Zeta(3))^(1/6) / (2^(1/3) * sqrt(3*Pi) * m^(1/12) * n^(2/3)).

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

Cf. A026007 (m=2), A263346 (m=3), A285262 (m=4).

Cf. A285246.

nmax = 40; CoefficientList[Series[Product[((1-x^(5*k))/(1-x^k))^k, {k,1,nmax}], {x,0,nmax}], x]

a(n) ~ exp(2^(1/3) * 3^(4/3) * 5^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * (2*Zeta(3))^(1/6) / (3^(1/3) * 5^(5/12) * 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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A285262 Expansion of Product_{k>=1} ((1-x^(4*k))/(1-x^k))^k.

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A285263 Expansion of Product_{k>=1} ((1-x^(5*k))/(1-x^k))^k.

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