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A256142
G.f.: Product_{j>=1} (1+x^j)^(3^j).
12
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
0,2
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, 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.
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... .
MATHEMATICA
nmax=30; CoefficientList[Series[Product[(1+x^k)^(3^k), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
Column k=3 of A292804.
Sequence in context: A180589 A370937 A042971 * A024038 A373960 A371429
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 16 2015
STATUS
approved