A256142 - OEIS

%I #9 Sep 24 2017 15:58:34

%S 1,3,12,55,225,927,3729,14787,57888,224220,860022,3270744,12343899,

%T 46264257,172305837,638039136,2350109736,8613851832,31428857611,

%U 114187160631,413222547846,1489829356657,5352683946903,19167988920930,68427472477338,243559693397025

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

%C 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)).

%H Vaclav Kotesovec, <a href="/A256142/b256142.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="/A034691/a034691_1.pdf">Asymptotics of sequence A034691</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

%F 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... .

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

%Y Cf. A102866, A144067, A144074.

%Y Column k=3 of A292804.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Mar 16 2015