A261567 Expansion of Product_{k>=1} (1/(1 + 3*x^k))^k.

1, -3, 3, -18, 69, -168, 504, -1578, 4800, -14310, 42396, -128049, 385839, -1154271, 3458847, -10386477, 31173873, -93490386, 280426833, -841384614, 2524300014, -7572585150, 22717270491, -68152872885, 204460229394, -613377236379, 1840126774737, -5520391488054

Offset: 0

Author

Vaclav Kotesovec, Aug 24 2015

Keywords

• sign

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. A255528, A261566, A261582.

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[(-1)^k*3^k/k*x^k/(1-x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

a(n) ~ c * (-3)^n, where c = Product_{j>=1} 1/(1 - 1/(-3)^j)^(j+1) = 0.72392917591300902192520561680114697538581509655711959502191898288595312452...