A293116 - OEIS

%I #34 Oct 04 2023 13:13:01

%S 1,-1,-1,-1,1,19,151,1091,7841,56519,396271,2442439,7701409,

%T -145269541,-4833158329,-104056218421,-2002667085119,-37109187217649,

%U -679877731030049,-12440309297451121,-227773259993414719,-4155839606711748061,-74724654677947488521

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

%H Seiichi Manyama, <a href="/A293116/b293116.txt">Table of n, a(n) for n = 0..450</a>

%H Geoffrey B. Campbell, <a href="https://arxiv.org/abs/2309.16094">Visible Point Vector Partition Identities for Hyperpyramid Lattices</a>, arXiv:2309.16094 [math.CO], 2023. See pp. 14, 27.

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

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

%F 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

%F 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

%p a:= proc(n) option remember; `if`(n=0, 1, -add(

%p       a(n-j)*binomial(n-1, j-1)*j!, j=1..n))

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 30 2017

%t CoefficientList[Series[E^(-x/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2017 *)

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

%Y Column k=0 of A293119.

%Y Cf. A000262, A066668, A111884.

%K sign

%O 0,6

%A _Seiichi Manyama_, Sep 30 2017