A303074 Expansion of Product_{n>=1} (1 + (9*x)^n)^(1/3).

1, 3, 18, 369, 1674, 31428, 266733, 3012714, 19924299, 319970007, 2688208641, 27248985549, 248061612240, 2597556114648, 25367004717831, 289880288735373, 2289952155529719, 23895509092285545, 252143223166599723, 2308267172943599733, 22389894059315522040

Offset: 0

Author

Vaclav Kotesovec, Apr 18 2018

Keywords

• nonn

Comments

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A271236, A298994, A303342, A370739.

Programs

Mathematica

CoefficientList[Series[(QPochhammer[-1, 9*x]/2)^(1/3), {x, 0, 20}], x]

Formula

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