A089148 Expansion of e.g.f.: 1/(exp(x) - x).

1, 0, -1, -1, 5, 19, -41, -519, -183, 19223, 73451, -847067, -8554547, 32488611, 977198559, 1325135969, -116987762287, -860498433233, 13730866757587, 243612350234973, -1120827248102379, -62079344419449925, -185852602587850681, 15185914155303053209

Offset: 0

Author

Wouter Meeussen, Dec 06 2003

Keywords

• sign

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A006153, A302397, A089087.

Programs

Maple

b:= proc(n) b(n):= -`if` (n<0, 1, add(b(n-i)/(i-1)!, i=1..n+1)) end:
a:= n-> (-1)^n*n!*b(n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 29 2013

Mathematica

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

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 */

Maxima

a(n):=n!*sum((-n-1+k)^k/k!,k,0,n); /* Tani Akinari, Mar 26 2023 */

PARI

x='x+O('x^66); Vec(serlaplace(1/(exp(x)-x))) \\ Joerg Arndt, May 29 2013

Sage

def A089148_list(len):
    f, R, C = 1, [], [1]+[0]*len
    for n in (1..len):
        for k in range(n, 0, -1):
            C[k] = C[k-1]*(1/(k-1) if k>1 else 1)
        C[0] = -sum((-1)^k*C[k] for k in (1..n))
        R.append(C[0]*f)
        f *= n
    return R
print(A089148_list(24)) # Peter Luschny, Feb 21 2016

Formula

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

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

Lim sup n->oo |a(n)/n!|^(1/n) = 1/abs(LambertW(-1)) = 0.727507111152... - Vaclav Kotesovec, Aug 13 2013

a(0) = 1; a(n) = -Sum_{k=2..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020

a(n) = n!*Sum_{k=0..n} (-n-1+k)^k/k!. - Tani Akinari, Mar 25 2023

a(n) = Sum_{k=0..n} A089087(n, k). - Peter Luschny, Mar 25 2023