A261652 Expansion of Product_{k>=0} ((1+x^(4*k+1))/(1-x^(4*k+1)))^3.

1, 6, 18, 38, 66, 108, 182, 306, 486, 728, 1068, 1578, 2318, 3312, 4614, 6388, 8862, 12192, 16488, 22038, 29400, 39156, 51702, 67554, 87810, 113982, 147384, 189200, 241446, 307356, 390408, 493662, 621006, 778712, 974628, 1216284, 1511756, 1872840, 2315538

Offset: 0

Author

Vaclav Kotesovec, Aug 28 2015

Keywords

• nonn

Comments

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} ((1 + x^(a*k+b))/(1 - x^(a*k+b)))^j, then a(n) ~ Gamma(b/a)^j * 2^(j/2 - 3/2 - 2*b*j/a) * a^(-j/4 - 1/4 + b*j/(2*a)) * exp(Pi*sqrt(j*n/a)) * j^(1/4 - j/4 + b*j/(2*a)) * Pi^(b*j/a - j) * n^(j/4 - 3/4 - b*j/(2*a)).

Crossrefs

Cf. A015128 (a=1, b=1, j=1), A156616.

Cf. A080054 (a=2, b=1, j=1), A007096 (a=2, b=1, j=2), A261647 (a=2, b=1, j=3), A014969 (a=2, b=1, j=4), A261648 (a=2, b=1, j=5), A014970 (a=2, b=1, j=6), A014972 (a=2, b=1, j=8), A103261 (a=2, b=1, j=10).

Cf. A261610 (a=3, b=1, j=1), A261649 (a=3, b=1, j=2), A261651 (a=3, b=1, j=3).

Cf. A261611 (a=4, b=1, j=1), A261650 (a=4, b=1, j=2), A261652 (a=4, b=1, j=3).

Programs

Mathematica

nmax=60; CoefficientList[Series[Product[((1+x^(4*k+1))/(1-x^(4*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ exp(Pi*sqrt(3*n)/2) * 2^(1/4) * Gamma(1/4)^3 / (8 * 3^(1/8) * Pi^(9/4) * n^(3/8)).