A263137 Expansion of Product_{k>=1} 1/(1-x^(4*k-3))^k.

1, 1, 1, 1, 1, 3, 3, 3, 3, 6, 9, 9, 9, 13, 19, 23, 23, 28, 42, 51, 56, 62, 84, 108, 120, 133, 170, 219, 253, 276, 335, 427, 503, 556, 650, 815, 977, 1090, 1244, 1525, 1836, 2079, 2344, 2808, 3386, 3876, 4348, 5107, 6121, 7069, 7932, 9176, 10918, 12671, 14257

Offset

0,6

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

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

Formula

G.f.: exp(Sum_{j>=1} 1/j*x^j/(1 - x^(4*j))^2).

a(n) ~ Zeta(3)^(29/288) * exp(d43 - Pi^4/(768*Zeta(3)) + Pi^2 * n^(1/3) / (16*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3)/4) / (2^(77/96) * sqrt(3*Pi) * n^(173/288)), where d43 = A263177 = Integral_{x=0..infinity} exp(-x)/(x*(1 - exp(-4*x))^2) - 1/(16*x^3) - 3/(16*x^2) - 19/(96*x*exp(x)) dx = 0.0960101036186695795680588847641594939129540181270663556962564550198... .

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+4, 4, 'r')=1, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100); # after Alois P. Heinz

Mathematica

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4k-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A263136, A263143, A263177, A263139.

Sequence in context: A280453 A289903 A285047 * A320846 A277515 A247202

Adjacent sequences: A263134 A263135 A263136 * A263138 A263139 A263140

Keyword

nonn

Author

Vaclav Kotesovec, Oct 10 2015

Status

approved