A294467 Binomial transform of A088311.

1, 2, 5, 22, 113, 746, 6037, 55070, 548417, 6281938, 79935941, 1087584422, 16109401585, 255667890362, 4358283982613, 79893373511086, 1542859916102657, 31322024816838050, 676027617881188357, 15287136167625123638, 362322855217463741681

Offset: 0

Author

Vaclav Kotesovec, Oct 31 2017

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..443

Crossrefs

Cf. A266232, A294467, A294468.

Cf. A218481, A294466, A281425, A095051.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)*(&*[1 + x^k: k in [1..50]]))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 15 2018

Mathematica

Table[Sum[Binomial[n, k]*k!*PartitionsQ[k], {k, 0, n}], {n, 0, 20}]

PARI

x='x+O('x^30); Vec(serlaplace(exp(x)*eta(x^2)/eta(x))) \\ G. C. Greubel, Oct 15 2018

Formula

a(n) ~ exp(1) * n! * A000009(n).

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

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