A278555 Expansion of Product_{n>=1} (1 - x^(5*n))^12/(1 - x^n)^13 in powers of x.

1, 13, 104, 637, 3276, 14808, 60541, 228124, 803010, 2667054, 8422715, 25446304, 73907808, 207209614, 562673618, 1484147681, 3811882087, 9553588317, 23407932874, 56161135485, 132132608899, 305240006266, 693150485885, 1548871015291, 3408852663762, 7395582677152

Offset: 0

Author

Seiichi Manyama, Nov 23 2016

Keywords

• nonn

Comments

In general, if m>0 and g.f. = Product_{k>=1} (1 - x^(5*k))^m/(1 - x^k)^(m+1) then a(n) ~ sqrt(4*m+5) * exp(Pi*sqrt(2*(4*m+5)*n/15)) / (4*sqrt(3)*5^((m+1)/2)*n). - Vaclav Kotesovec, Nov 24 2016

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2500
• 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.

Crossrefs

Cf. A160460, A278556, A278557, A278558, A278559.

Programs

Mathematica

CoefficientList[ Series[ Product[(1 - x^(5n))^12/(1 - x^n)^13, {n, 25}],
{x, 0, 25}], x] (* Robert G. Wilson v, Nov 23 2016 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))^12/(1 - x^n)^13.

A278559(n) = 5^2*63*A160460(n) + 5^5*52*a(n-1) + 5^7*63*A278556(n-2) + 5^10*6*A278557(n-3) + 5^12*A278558(n-4) for n >= 4.

a(n) ~ sqrt(53/15)*exp(sqrt(106*n/15)*Pi)/(62500*n). - Vaclav Kotesovec, Nov 24 2016