A294468 Inverse binomial transform of A088311.

1, 0, 1, 8, 9, 224, 1225, 11304, 103537, 1431296, 15642801, 206721800, 3295533241, 47467875168, 859354139449, 15596241280424, 283240963555425, 5859309797252864, 129874369387025377, 2752905169704533256, 67640333903657850601

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, A293467.

Cf. A218481, A294466, A281425, A095051.

Programs

Mathematica

Table[Sum[(-1)^(n-k)*Binomial[n, k]*k!*PartitionsQ[k], {k, 0, n}], {n, 0, 20}]
max = 20; t = Table[k!*PartitionsQ[k], {k, 0, max}]; Table[Differences[t, n], {n, 0, max}][[All, 1]] (* Jean-François Alcover, Nov 02 2017 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A088311(k).

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