A303381 Expansion of Product_{n>=1} ((1 + (8*x)^n)/(1 - (8*x)^n))^(1/8).

1, 2, 18, 204, 1526, 15228, 146676, 1217880, 10322982, 106429420, 886934236, 7632390312, 72137002428, 600860144728, 5351962341672, 51402944345520, 411439139563526, 3624067316629836, 33666668386023244, 279519776297893512, 2480351338204454484

Offset: 0

Author

Seiichi Manyama, Apr 22 2018

Keywords

• nonn

Comments

In general, if h>=1 and g.f. = Product_{k>=1} ((1 + (h*x)^k)/(1 - (h*x)^k))^(1/h), then a(n) ~ h^n * exp(Pi*sqrt(n/h)) /(2^(3/2 + 3/(2*h)) * h^(1/4 + 1/(4*h)) * n^(3/4 + 1/(4*h))). - Vaclav Kotesovec, Apr 23 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Expansion of Product_{n>=1} ((1 + (2^b*x)^n)/(1 - (2^b)*x^n))^(1/(2^b)): A015128 (b=0), A303307 (b=1), A303361 (b=2), this sequence (b=3).

Cf. A303382.

Programs

Maple

seq(coeff(series(mul(((1+(8*x)^k)/(1-(8*x)^k))^(1/8), k = 1..n), x, n+1), x, n), n = 0..25); # Muniru A Asiru, Apr 23 2018

Mathematica

nmax = 20; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(1/8), {k, 1, nmax}], {x, 0, nmax}], x] * 8^Range[0, nmax] (* Vaclav Kotesovec, Apr 23 2018 *)

PARI

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+(8*x)^k)/(1-(8*x)^k))^(1/8)))

Formula

a(n) ~ 2^(3*n - 81/32) * exp(sqrt(n)*Pi/2^(3/2)) / n^(25/32). - Vaclav Kotesovec, Apr 23 2018