A266108 Expansion of Product_{k>=1} (1 + (x + x^2)^k).

1, 1, 2, 4, 9, 17, 33, 67, 133, 256, 488, 935, 1798, 3437, 6507, 12239, 22950, 42959, 80283, 149717, 278338, 515579, 952014, 1753899, 3225529, 5921773, 10852501, 19853341, 36254081, 66082021, 120233759, 218396940, 396114374, 717473628, 1297869159, 2344798633

Offset: 0

Author

Vaclav Kotesovec, Dec 21 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..4550

Crossrefs

Cf. A238441, A266124.

Programs

Mathematica

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

Formula

a(n) = Sum_{k=0..n} binomial(k,n-k)*q(k), where q(n) is A000009.

a(n) ~ phi^(n+1/4) * exp(Pi*sqrt(phi*n/(3*sqrt(5))) + Pi^2/(120*phi)) / (4*3^(1/4)*5^(1/8)*n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.