A256142 G.f.: Product_{j>=1} (1+x^j)^(3^j).

1, 3, 12, 55, 225, 927, 3729, 14787, 57888, 224220, 860022, 3270744, 12343899, 46264257, 172305837, 638039136, 2350109736, 8613851832, 31428857611, 114187160631, 413222547846, 1489829356657, 5352683946903, 19167988920930, 68427472477338, 243559693397025

Offset: 0

Author

Vaclav Kotesovec, Mar 16 2015

Keywords

• nonn

Comments

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Asymptotics of sequence A034691
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Crossrefs

Cf. A102866, A144067, A144074.

Column k=3 of A292804.

Programs

Mathematica

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

Formula

a(n) ~ 3^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(3^(m-1)-1)) = 0.215985336303958581708278160877115129... .