A261611 Expansion of Product_{k>=0} (1 + x^(4*k+1))/(1 - x^(4*k+1)).

1, 2, 2, 2, 2, 4, 6, 6, 6, 8, 12, 14, 14, 16, 22, 28, 30, 32, 40, 50, 56, 60, 70, 86, 98, 106, 120, 144, 166, 180, 200, 234, 270, 296, 324, 372, 428, 472, 514, 580, 664, 736, 800, 890, 1010, 1124, 1222, 1346, 1514, 1684, 1834, 2008, 2240, 2488, 2712, 2956

Offset: 0

Author

Vaclav Kotesovec, Aug 26 2015

Keywords

• nonn

Comments

In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k+b))/(1 - x^(a*k+b)), then a(n) ~ Gamma(b/a) * a^(b/(2*a) - 1/2) * Pi^(b/a - 1) * exp(Pi*sqrt(n/a)) / (2^(2*b/a + 1) * n^(b/(2*a) + 1/2)).

Crossrefs

Cf. A015128, A080054, A261610.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Product[(1 + x^(4*k+1))/(1 - x^(4*k+1)), {k, 0, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(n)/2) * Gamma(1/4) / (2^(9/4) * Pi^(3/4) * n^(5/8)).