A089148 - OEIS

%I #59 Mar 28 2023 23:03:19

%S 1,0,-1,-1,5,19,-41,-519,-183,19223,73451,-847067,-8554547,32488611,

%T 977198559,1325135969,-116987762287,-860498433233,13730866757587,

%U 243612350234973,-1120827248102379,-62079344419449925,-185852602587850681,15185914155303053209

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

%C INVERTi transform of [1, 1, 1/2, 1/6, 1/24, 1/120, ...] = [1, 0, -1/2, 1/6, 5/24, -19/120, -41/720, 519/5040, -183/40320, -19223/362880, ...]. - _Gary W. Adamson_, Oct 08 2008

%H Alois P. Heinz, <a href="/A089148/b089148.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: -(1+1/(G(0)-1))/x where G(k) = 1 - (k+1)/(1 - x/(x + (k+1)^2/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 07 2012

%F a(n) = Sum_{k=0..n} Sum_{m=0..k} k!*binomial(n,m)*(-1)^(k-m)*Stirling2(n-m,k-m). - _Vladimir Kruchinin_, May 29 2013

%F Lim sup n->oo |a(n)/n!|^(1/n) = 1/abs(LambertW(-1)) = 0.727507111152... - _Vaclav Kotesovec_, Aug 13 2013

%F a(0) = 1; a(n) = -Sum_{k=2..n} binomial(n,k) * a(n-k). - _Ilya Gutkovskiy_, Feb 09 2020

%F a(n) = n!*Sum_{k=0..n} (-n-1+k)^k/k!. - _Tani Akinari_, Mar 25 2023

%F a(n) = Sum_{k=0..n} A089087(n, k). - _Peter Luschny_, Mar 25 2023

%p b:= proc(n) b(n):= -`if` (n<0, 1, add(b(n-i)/(i-1)!, i=1..n+1)) end:

%p a:= n-> (-1)^n*n!*b(n):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, May 29 2013

%t a = CoefficientList[Series[1/( E^ x - x), {x, 0, 30}], x]; Table[(n - 1)! *a[[n]], {n, 1, Length[a]}] (* _Roger L. Bagula_ and _Gary W. Adamson_, Oct 06 2008 *)

%t With[{nn=30},CoefficientList[Series[1/(Exp[x]-x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 03 2017 *)

%o (Maxima) a(n):=sum(sum(k!*binomial(n,l)*(-1)^(k-l)*stirling2(n-l,k-l), l,0,k), k,0,n); /* _Vladimir Kruchinin_, May 29 2013 */

%o (Maxima) a(n):=n!*sum((-n-1+k)^k/k!,k,0,n); /* _Tani Akinari_, Mar 26 2023 */

%o (PARI) x='x+O('x^66); Vec(serlaplace(1/(exp(x)-x))) \\ _Joerg Arndt_, May 29 2013

%o (Sage)

%o def A089148_list(len):

%o     f, R, C = 1, [], [1]+[0]*len

%o     for n in (1..len):

%o         for k in range(n, 0, -1):

%o             C[k] = C[k-1]*(1/(k-1) if k>1 else 1)

%o         C[0] = -sum((-1)^k*C[k] for k in (1..n))

%o         R.append(C[0]*f)

%o         f *= n

%o     return R

%o print(A089148_list(24)) # _Peter Luschny_, Feb 21 2016

%Y Cf. A006153, A302397, A089087.

%K sign

%O 0,5

%A _Wouter Meeussen_, Dec 06 2003