A303342 Expansion of Product_{k>=1} ((1 + (9*x)^k) / (1 - (9*x)^k))^(1/3).

1, 6, 72, 1008, 10746, 130896, 1569456, 17371584, 192625128, 2260005462, 24725148912, 270748885392, 3027318848208, 32608207056528, 354309508944288, 3902606972751168, 41393526342215994, 443390745816982944, 4783687280410092984, 50532141192366275280

Offset: 0

Author

Vaclav Kotesovec, Apr 22 2018

Keywords

• nonn

Comments

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

Crossrefs

Cf. A271236, A303074, A303307.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Product[((1+(9*x)^k)/(1-(9*x)^k))^(1/3), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 3^(2*n) * exp(Pi*sqrt(n/3)) / (4 * 3^(1/3) * n^(5/6)).