A300579 Expansion of Product_{k>=1} 1/(1 - 3^(k-1)*x^k).

1, 1, 4, 13, 49, 157, 589, 1885, 6826, 22378, 78754, 256630, 904711, 2934247, 10133851, 33287620, 113522089, 370582069, 1262300701, 4110883510, 13869616495, 45364050184, 151708228636, 494743296757, 1654133919475, 5379427446952, 17858926956532, 58219580395822

Offset: 0

Author

Vaclav Kotesovec, Mar 09 2018

Keywords

• nonn

Comments

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A075900, A338673, A338674, A338675, A338676, A338677, A338678, A338679.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[1/(1 - 3^(k-1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ polylog(2, 1/3)^(1/4) * 3^(n - 1/2) * exp(2*sqrt(polylog(2, 1/3)*n)) / (sqrt(2*Pi) * n^(3/4)), where polylog(2, 1/3) = 0.36621322997706348761674629...

a(n) = Sum_{k=0..n} p(n,k) * 3^(n-k), where p(n,k) is the number of partitions of n into k parts. - Ilya Gutkovskiy, Jun 08 2022