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

0,2

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.

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

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 *)

Crossrefs

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

Sequence in context: A289859 A129762 A283121 * A282921 A023011 A022641

Adjacent sequences: A278552 A278553 A278554 * A278556 A278557 A278558

Keyword

nonn

Author

Seiichi Manyama, Nov 23 2016

Status

approved