A238441 Expansion of 1/Product_{n>=1} (1 - (q + q^2)^n).

1, 1, 3, 7, 16, 36, 79, 171, 367, 776, 1625, 3379, 6969, 14262, 29001, 58644, 117951, 235994, 469904, 931642, 1839708, 3618893, 7092676, 13853271, 26970933, 52350092, 101316743, 195544281, 376411466, 722747148, 1384416306, 2645765058, 5045240163, 9600533209, 18231674112, 34554871809, 65369632350, 123440337791

Offset

0,3

Comments

What does this sequence count?

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..4400

Formula

G.f.: 1/Product_{n>=1} (1 - (q + q^2)^n).

G.f.: P(x+x^2), where P(x) is g.f. of A000041, a(n) = Sum_{k=0..n} binomial(k,n-k)*p(k), where p(n) is number of partitions n. - Vladimir Kruchinin, Dec 21 2015

a(n) ~ phi^n * exp(Pi*sqrt(2*phi*n/(3*sqrt(5))) + Pi^2/(60*phi)) / (4*sqrt(3)*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 21 2015

Mathematica

nmax=40; CoefficientList[Series[Product[1/(1 - (x+x^2)^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 21 2015 *)

Prog

(PARI) q = 'q + O('q^66); Vec(1/eta(q*(1+q)))

Crossrefs

Cf. A266108, A266124.

Sequence in context: A023523 A065979 A106463 * A173514 A045891 A081037

Adjacent sequences: A238438 A238439 A238440 * A238442 A238443 A238444

Keyword

nonn

Author

Joerg Arndt, Feb 27 2014

Status

approved