%I #22 Nov 24 2016 09:47:56

%S 1,13,104,637,3276,14808,60541,228124,803010,2667054,8422715,25446304,

%T 73907808,207209614,562673618,1484147681,3811882087,9553588317,

%U 23407932874,56161135485,132132608899,305240006266,693150485885,1548871015291,3408852663762,7395582677152

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

%C 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

%H Vaclav Kotesovec, <a href="/A278555/b278555.txt">Table of n, a(n) for n = 0..2500</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015.

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

%F 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.

%F a(n) ~ sqrt(53/15)*exp(sqrt(106*n/15)*Pi)/(62500*n). - _Vaclav Kotesovec_, Nov 24 2016

%t CoefficientList[ Series[ Product[(1 - x^(5n))^12/(1 - x^n)^13, {n, 25}],

%t {x, 0, 25}], x] (* _Robert G. Wilson v_, Nov 23 2016 *)

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

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 23 2016