OFFSET
0,4
REFERENCES
N. G. de Bruijn, Asymptotic Methods in Analysis, Dover Publications, 1981, p. 24.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
E.g.f.: -LambertW(-x*exp(-2*x))/x. - Vladeta Jovovic, Aug 25 2004
a(n) = Sum_{k=0..n} binomial(n,k)*((k+1)^(n-1)*2^(n-k)*(-1)^(n+k)). - Vladimir Kruchinin, Jan 31 2012
Lim sup_{n->infinity} (|a(n)|/n!)^(1/n) = 2/abs(LambertW(-2*exp(-1))) = 1.598960348180173... - Vaclav Kotesovec, Jul 26 2013
MATHEMATICA
max = 19; (Series[ -ProductLog[ -Exp[-2*x]*x]/x, {x, 0, max}] // CoefficientList[#, x] &) * Range[0, max]! (* Jean-François Alcover, Jun 24 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Nest[ Exp[ (#-2) x]&, 1 + O[x], n], {x, 0, n]]]; (* Michael Somos, Jun 17 2018 *)
PROG
(PARI) {a(n) = my(A); if(n<0, 0, A = 1 + O(x); for(k=1, n, A = exp(x*A - 2*x)); n! * polcoeff(A, n))};
(Maxima) a(n):=sum(binomial(n, k)*((k+1)^(n-1)*2^(n-k)*(-1)^(n+k)), k, 0, n); /* Vladimir Kruchinin, Jan 31 2012 */
(GAP) List([0..20], n->Sum([0..n], k->Binomial(n, k)*((k+1)^(n-1)*2^(n-k)*(-1)^(n+k)))); # Muniru A Asiru, Jun 17 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 22 2004
EXTENSIONS
Typo in third formula corrected by Vaclav Kotesovec, Feb 17 2014
STATUS
approved