A347830 a(n) = Sum_{k=0..n} 2^k * A000009(k) * A000041(n-k).

1, 3, 8, 27, 67, 189, 509, 1329, 3344, 8694, 22062, 54756, 136741, 335103, 822277, 2016738, 4872787, 11711655, 28253743, 67319328, 160333627, 381350646, 901272326, 2121969771, 4991176893, 11689645776, 27305992220, 63705989106, 148106539514, 343371565449, 795524336390

Offset: 0

Author

Vaclav Kotesovec, Sep 15 2021

Keywords

• nonn

Crossrefs

Cf. A000009, A000041, A015128, A065446, A264686, A347829.

Programs

Mathematica

Table[Sum[2^k*PartitionsQ[k]*PartitionsP[n-k], {k, 0, n}], {n, 0, 50}]
nmax = 50; CoefficientList[Series[Product[(1 + 2^k*x^k) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ A065446 * 2^n * A000009(n).

a(n) ~ 2^(n-2) * exp(Pi*sqrt(n/3)) / (3^(1/4) * QPochhammer(1/2) * n^(3/4)).

G.f.: Product_{k>=1} (1 + 2^k*x^k) / (1 - x^k).