A346964 Expansion of e.g.f. Product_{k>=1} exp(x^k) * (1 + x^k).

1, 1, 2, 7, 40, 257, 2086, 19567, 207572, 2451745, 32226922, 462314711, 7178502112, 120315808417, 2157566463950, 41277697722367, 838883560646476, 18020304830796737, 408135672764386642, 9723868266912217255, 242827969365094823192, 6345340713682009241281

Offset: 0

Author

Vaclav Kotesovec, Aug 09 2021

Keywords

• nonn

Crossrefs

Cf. A000262, A088311, A330200.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Product[Exp[x^k] * (1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[x/(1 - x)] * Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[Product[(1 + x^k) / (1 - x^k)^(EulerPhi[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
Table[n!*Sum[LaguerreL[k, -1, -1]*PartitionsQ[n-k],{k,0,n}], {n,0,20}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A000262(k) * A088311(n-k).

E.g.f.: Product_{k>=1} (1 + x^k) / (1 - x^k)^(A000010(k)/k).

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