A303381 - OEIS

%I #18 Apr 24 2018 02:22:42

%S 1,2,18,204,1526,15228,146676,1217880,10322982,106429420,886934236,

%T 7632390312,72137002428,600860144728,5351962341672,51402944345520,

%U 411439139563526,3624067316629836,33666668386023244,279519776297893512,2480351338204454484

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

%C 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

%H Seiichi Manyama, <a href="/A303381/b303381.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ 2^(3*n - 81/32) * exp(sqrt(n)*Pi/2^(3/2)) / n^(25/32). - _Vaclav Kotesovec_, Apr 23 2018

%p 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

%t 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 *)

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

%Y 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).

%Y Cf. A303382.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 22 2018