OFFSET
0,5
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
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
Wouter Meeussen, Dec 06 2003
STATUS
approved