A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)*x^k) * (1 + 3^(k-1)*x^k).

1, 10, 39, 390, 1521, 7830, 49518, 207360, 951102, 4264650, 22185657, 89579520, 401428224, 1676401110, 7172977275, 31972081050, 130330236546, 537393139200, 2213787635712, 8988968449530, 36073295687070, 150459195064320, 590262148332288, 2362876271009370, 9314694641056095

Offset: 0

Author

Vaclav Kotesovec, Mar 02 2024

Keywords

• nonn

Comments

In general, if d >= 1 and g.f. = Product_{k>=1} (1 + d^(k+1)*x^k) * (1 + d^(k-1)*x^k), then a(n) ~ d^(n + 1/2) * exp(sqrt(2*n*(Pi^2/3 + log(d)^2))) * (Pi^2/3 + log(d)^2)^(1/4) / (2^(5/4) * sqrt(Pi) * (d+1) * n^(3/4)).

Crossrefs

Cf. A032308, A344062.

Cf. A022567 (d=1), A370761 (d=2).

Programs

Mathematica

nmax = 25; CoefficientList[Series[Product[(1+3^(k+1)*x^k)*(1+3^(k-1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 3^(n + 1/2) * exp(sqrt(2*n*(Pi^2/3 + log(3)^2))) * (Pi^2/3 + log(3)^2)^(1/4) / (2^(13/4) * sqrt(Pi) * n^(3/4)).