A266124 Expansion of Product_{k>=1} (1 + (x+x^2)^k) / (1 - (x+x^2)^k).

1, 2, 6, 16, 42, 104, 252, 600, 1402, 3218, 7282, 16288, 36048, 78988, 171516, 369416, 789762, 1676818, 3537622, 7419544, 15475756, 32112968, 66313088, 136312608, 279000612, 568732738, 1154881834, 2336565080, 4710930856, 9466623964, 18963077484, 37871190504

Offset

0,2

Comments

Convolution of A266108 and A238441.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Formula

a(n) ~ phi^n * exp(Pi*5^(-1/4)*sqrt(phi*n) + Pi^2/(40*phi)) / (8*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

Mathematica

nmax=40; CoefficientList[Series[Product[(1+(x+x^2)^k)/(1-(x+x^2)^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Binomial[k, n-k] * Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}], {k, 0, n}], {n, 0, 40}]

Crossrefs

Cf. A266108, A238441.

Sequence in context: A365548 A143123 A102699 * A217194 A304662 A296625

Adjacent sequences: A266121 A266122 A266123 * A266125 A266126 A266127

Keyword

nonn

Author

Vaclav Kotesovec, Dec 21 2015

Status

approved