A261565 Expansion of Product_{k>=1} (1/(1 - 3*x^k))^k.

1, 3, 15, 54, 201, 672, 2268, 7266, 23208, 72414, 224652, 688929, 2103975, 6386907, 19337091, 58367817, 175905741, 529331190, 1591515297, 4781575074, 14359673454, 43108645230, 129387584991, 388283978589, 1165099808574, 3495782937135, 10488322595625

Offset: 0

Author

Vaclav Kotesovec, Aug 24 2015

Keywords

• nonn

Comments

In general, for z > 1 or z < -1, if g.f. = Product_{k>=1} (1/(1 - z*x^k))^k, then a(n) ~ c * z^n, where c = Product_{j>=1} 1/(1 - 1/z^j)^(j+1).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000219, A242587, A246935, A261561.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[(1/(1 - 3*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[3^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

PARI

{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, 3^d * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015

Formula

a(n) ~ c * 3^n, where c = Product_{j>=1} 1/(1 - 1/3^j)^(j+1) = 4.1269357592430271005054028580646705856298720432004233223482475759761040273...

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} 3^d * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015