A266820 Expansion of Product_{k>=1} ((1 + 2*x^k) * (1 + 3*x^k)).

1, 5, 11, 30, 66, 115, 252, 445, 762, 1350, 2238, 3690, 5909, 9480, 14460, 22475, 34326, 51150, 76398, 111810, 163350, 236610, 339667, 482040, 684060, 960780, 1340953, 1863570, 2573022, 3533310, 4830822, 6580170, 8900382, 12011430, 16125198, 21567965

Offset

0,2

Comments

Convolution of A032302 and A032308.

In general, for m1 > 0 and m2 > 0, if g.f. = Product_{k>=1} ((1 + m1*x^k) * (1 + m2*x^k)) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m1+1)*(m2+1)*Pi) * n^(3/4)), where c = Pi^2/3 + log(m1)^2/2 + log(m2)^2/2 + polylog(2, -1/m1) + polylog(2, -1/m2).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(3*Pi) * n^(3/4)), where c = Pi^2/3 + log(2)^2/2 + log(3)^2/2 + polylog(2, -1/2) + polylog(2, -1/3) = 6.665989921346842772385004076363525173910446415877... .

Mathematica

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

Crossrefs

Cf. A032302, A032308, A266819, A266822.

Sequence in context: A088484 A179256 A209659 * A335455 A114688 A257717

Adjacent sequences: A266817 A266818 A266819 * A266821 A266822 A266823

Keyword

nonn

Author

Vaclav Kotesovec, Jan 04 2016

Status

approved