A285132 - OEIS

%I #25 Apr 24 2017 12:51:17

%S 1,0,0,0,4,0,0,0,10,9,0,0,20,36,14,0,35,90,101,19,56,180,320,202,108,

%T 315,730,859,492,533,1390,2300,2139,1354,2393,4835,6475,5098,4619,

%U 8813,14926,16395,12982,15751,28962,41162,40256,35200,51731,85365,106145

%N Expansion of Product_{k>=0} 1/(1-x^(5*k+4))^(5*k+4).

%C In general, if m > 1 and g.f. = Product_{k>=1} 1/(1-x^(m*k-1))^(m*k-1), then a(n, m) ~ exp(c*m + 3 * 2^(-2/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * (2*Zeta(3))^(1/(6*m) + m/36) / (sqrt(3) * Gamma(1 - 1/m) * m^(1/2 - 5/(6*m) + m/36) * n^(1/2 + 1/(6*m) + m/36)), where c = Integral_{x=0..infinity} exp((m+1)*x) / (x*(exp(m*x)-1)^2) + (1/12 - 1/(2*m^2))/(x*exp(x)) - 1/(m^2*x^3) - 1/(m^2*x^2) dx. - _Vaclav Kotesovec_, Apr 17 2017

%H Seiichi Manyama, <a href="/A285132/b285132.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ exp(5*c + 3*2^(-2/3)*5^(-1/3)*Zeta(3)^(1/3)*n^(2/3)) * (2*Zeta(3))^(31/180) / (sqrt(3) * 5^(17/36) * Gamma(4/5) * n^(121/180)), where c = Integral_{x=0..inf} ((19/(exp(x)*300) + 1/(exp(4*x)*(1-exp(-5*x))^2) - 1/(25*x^2) - 1/(25*x))/x) dx = -0.12699586713882325294527057473113580561183418857868946729897216431919... - _Vaclav Kotesovec_, Apr 15 2017

%t nmax = 100; CoefficientList[Series[Product[1/(1-x^(5*k-1))^(5*k-1), {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Apr 15 2017 *)

%o (PARI) x='x+O('x^100); Vec(prod(k=0, 100, 1/(1 - x^(5*k + 4))^(5*k + 4))) \\ _Indranil Ghosh_, Apr 15 2017

%Y Product_{k>=0} 1/(1-x^(m*k+m-1))^(m*k+m-1): A262811 (m=2), A262946 (m=3), A285131 (m=4), this sequence (m=5).

%Y Cf. A285214.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Apr 15 2017