A266857 Expansion of Product_{k>=1} (1 + 3*x^k)^k.

1, 3, 6, 27, 48, 132, 324, 651, 1491, 3078, 6447, 12795, 25839, 50088, 96099, 184491, 343920, 640545, 1173609, 2138403, 3850584, 6882354, 12186336, 21423660, 37421757, 64816608, 111637392, 190976859, 324868530, 549265290, 923904711, 1545406077, 2572326510

Offset

0,2

Comments

In general, for m > 0, if g.f. = Product_{k>=1} (1 + m*x^k)^k then a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (3^(2/3) * (m+1)^(1/12) * sqrt(2*Pi) * n^(2/3)), where c = Pi^2*log(m) + log(m)^3 - 6*polylog(3, -1/m).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Formula

a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (2^(2/3) * 3^(2/3) * sqrt(Pi) * n^(2/3)), where c = Pi^2*log(3) + log(3)^3 - 6*polylog(3, -1/3) = 14.092743327504459346835224018840792668682349056875722467... .

Mathematica

nmax=50; CoefficientList[Series[Product[(1+3*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A026007, A261562, A261565, A261567.

Sequence in context: A208665 A256762 A064283 * A014561 A034502 A217725

Adjacent sequences: A266854 A266855 A266856 * A266858 A266859 A266860

Keyword

nonn

Author

Vaclav Kotesovec, Jan 04 2016

Status

approved