A294466 Binomial transform of A053529.

1, 2, 7, 34, 221, 1666, 15187, 153602, 1770169, 22379266, 312164831, 4685997922, 76668261397, 1335425319554, 24921410400811, 493075754663746, 10358312736025457, 228862423291312642, 5335861084579488439, 130235118120543955106, 3333808742649699747661

Offset: 0

Author

Vaclav Kotesovec, Oct 31 2017

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..441

Crossrefs

Cf. A218481, A281425, A095051.

Cf. A266232, A294467, A293467, A294468.

Programs

Mathematica

Table[Sum[Binomial[n, k]*k!*PartitionsP[k], {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Exp[x] * x^(1/24)/DedekindEta[Log[x]/(2*Pi*I)], {x, 0, nmax}], x] * Range[0, nmax]!

PARI

x='x+O('x^50); Vec(serlaplace(exp(x)/eta(x))) \\ G. C. Greubel, Oct 15 2018

Formula

E.g.f.: exp(x)/eta(x), where eta(x) is the Dedekind eta function.

a(n) ~ exp(1) * n! * A000041(n).

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

E.g.f.: exp(x + Sum_{k>=1} sigma(k)*x^k/k). - Ilya Gutkovskiy, Oct 15 2018