A285132 Expansion of Product_{k>=0} 1/(1-x^(5*k+4))^(5*k+4).

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, 315, 730, 859, 492, 533, 1390, 2300, 2139, 1354, 2393, 4835, 6475, 5098, 4619, 8813, 14926, 16395, 12982, 15751, 28962, 41162, 40256, 35200, 51731, 85365, 106145

Offset: 0

Author

Seiichi Manyama, Apr 15 2017

Keywords

• nonn

Comments

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

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).

Cf. A285214.

Programs

Mathematica

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 *)

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

Formula

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