A285049 Expansion of Product_{k>=0} 1/(1-x^(5*k+1))^(5*k+1).

1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 7, 18, 39, 39, 39, 39, 55, 121, 177, 177, 177, 198, 360, 591, 717, 717, 743, 1045, 1777, 2393, 2645, 2676, 3199, 4982, 7264, 8650, 9148, 9956, 13760, 20348, 26060, 28873, 30869, 38134, 54634, 73142, 85536, 92302, 106501, 143167

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-m+1))^(m*k-m+1), then a(n, m) ~ exp(c*m + 3 * 2^(-2/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(19*m/36 + 1/(6*m) - 1) * m^(17*m/36 + 5/(6*m) - 3/2) * Pi^(m/2 - 1) * Zeta(3)^(1/(6*m) + m/36) / (sqrt(3) * Gamma(1/m)^(m-1) * n^(1/2 + 1/(6*m) + m/36)), where c = Integral_{x=0..infinity} exp((2*m-1)*x) / (x*(exp(m*x) - 1)^2) + (1/12 - (m-1)^2/(2*m^2))/(x*exp(x)) - 1/(m^2*x^3) - (m-1)/(m^2*x^2) dx. - Vaclav Kotesovec, Apr 17 2017

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Crossrefs

Product_{k>=0} 1/(1-x^(m*k+1))^(m*k+1): A000219 (m=1), A262811 (m=2), A262947 (m=3), A285048 (m=4), this sequence (m=5).

Cf. A285071.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[1/(1-x^(5*k-4))^(5*k-4), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

Formula

a(n) ~ 2^(301/180) * 5^(37/36) * Pi^(3/2) * Zeta(3)^(31/180) * exp(5*c + 3 * 2^(-2/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) / (sqrt(3) * Gamma(1/5)^4 * n^(121/180)), where c = Integral_{x=0..inf} ((-71/(exp(x)*300) + 1/(exp(x)*(1 - exp(-5*x))^2) - 1/(25*x^2) - 4/(25*x))/x) dx = 0.186382690624752630391368364629918483384424086341764409146923686... - Vaclav Kotesovec, Apr 16 2017