A097718 E.g.f. A(x) satisfies A(x) = exp(x(A(x)-2)).

1, -1, -1, 2, 21, 54, -605, -8422, -17815, 915470, 13791711, -14182158, -3814159811, -55759417546, 472583147387, 33181980839114, 418144112565969, -10448831982433506, -511822958265199817, -4431070683610565086

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

Sequence in context: A074875 A231807 A351118 * A180232 A075681 A212257

Adjacent sequences: A097715 A097716 A097717 * A097719 A097720 A097721

Keyword

sign

Author

Michael Somos, Aug 22 2004

Extensions

Typo in third formula corrected by Vaclav Kotesovec, Feb 17 2014

Status

approved