A293116 Expansion of e.g.f. exp(x/(x-1)).

1, -1, -1, -1, 1, 19, 151, 1091, 7841, 56519, 396271, 2442439, 7701409, -145269541, -4833158329, -104056218421, -2002667085119, -37109187217649, -679877731030049, -12440309297451121, -227773259993414719, -4155839606711748061, -74724654677947488521

Offset: 0

Author

Seiichi Manyama, Sep 30 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..450
• Geoffrey B. Campbell, Visible Point Vector Partition Identities for Hyperpyramid Lattices, arXiv:2309.16094 [math.CO], 2023. See pp. 14, 27.

Crossrefs

Column k=0 of A293119.

Cf. A000262, A066668, A111884.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, -add(
      a(n-j)*binomial(n-1, j-1)*j!, j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 30 2017

Mathematica

CoefficientList[Series[E^(-x/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

PARI

my(x='x+O('x^66)); Vec(serlaplace(exp(x/(x-1))))

Formula

E.g.f.: exp(x/(x-1)).

a(n) = (-1)^n * A111884(n).

E.g.f.: Product_{k>=1} (1 - x^k)^(phi(k)/k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, May 25 2019

D-finite with recurrence a(n) +(-2*n+3)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Mar 13 2023