A285674 Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^k.

1, 1, 9, 36, 148, 489, 1959, 6326, 22741, 74072, 246436, 781189, 2523042, 7773342, 24200874, 73439472, 222247101, 660405663, 1958564056, 5715567301, 16623600991, 47780474694, 136623175876, 386983158080, 1090779014163, 3048348195528, 8478106666045

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Formula

a(n) ~ c * 3^(2*n/3) * n^2, where

c = 76631915822.1860553820452485980060616094557062528483009... if mod(n,3)=0

c = 76631915822.1819974623120987784506295282600132985390786... if mod(n,3)=1

c = 76631915822.1825610530012010285873110459423711856434442... if mod(n,3)=2

In closed form, a(n) ~ (Product_{k>=4}((1 - k^2/3^(2*k/3))^(-k)) / ((1 - 1/3^(2/3)) * (1 - 4/3^(4/3))^2) + Product_{k>=4}((1 - (-1)^(2*k/3)*k^2/3^(2*k/3))^(-k)) / ((-1)^(2*n/3) * ((1 + 4/3*(-1/3)^(1/3))^2 * (1 - (-1/3)^(2/3)))) + Product_{k>=4}((1 - (-1)^(4*k/3)*k^2/3^(2*k/3))^(-k)) / ((-1)^(4*n/3) * ((1 + (-1)^(1/3)/3^(2/3)) * (1 - 4*(-1)^(2/3) / 3^(4/3))^2))) * 3^(2*n/3) * n^2 / 54.

Mathematica

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

Crossrefs

Cf. A077335, A266941, A285241, A285737.

Sequence in context: A231431 A264515 A002063 * A075674 A245416 A265837

Adjacent sequences: A285671 A285672 A285673 * A285675 A285676 A285677

Keyword

nonn

Author

Vaclav Kotesovec, Apr 24 2017

Status

approved