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%I #35 Jun 21 2018 02:20:25
%S 1,-1,-1,2,21,54,-605,-8422,-17815,915470,13791711,-14182158,
%T -3814159811,-55759417546,472583147387,33181980839114,418144112565969,
%U -10448831982433506,-511822958265199817,-4431070683610565086
%N E.g.f. A(x) satisfies A(x) = exp(x(A(x)-2)).
%D N. G. de Bruijn, Asymptotic Methods in Analysis, Dover Publications, 1981, p. 24.
%H Vincenzo Librandi, <a href="/A097718/b097718.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: -LambertW(-x*exp(-2*x))/x. - _Vladeta Jovovic_, Aug 25 2004
%F a(n) = Sum_{k=0..n} binomial(n,k)*((k+1)^(n-1)*2^(n-k)*(-1)^(n+k)). - _Vladimir Kruchinin_, Jan 31 2012
%F Lim sup_{n->infinity} (|a(n)|/n!)^(1/n) = 2/abs(LambertW(-2*exp(-1))) = 1.598960348180173... - _Vaclav Kotesovec_, Jul 26 2013
%t max = 19; (Series[ -ProductLog[ -Exp[-2*x]*x]/x, {x, 0, max}] // CoefficientList[#, x] &) * Range[0, max]! (* _Jean-François Alcover_, Jun 24 2013 *)
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Nest[ Exp[ (#-2) x]&, 1 + O[x], n], {x, 0, n]]]; (* _Michael Somos_, Jun 17 2018 *)
%o (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))};
%o (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 */
%o (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
%K sign
%O 0,4
%A _Michael Somos_, Aug 22 2004
%E Typo in third formula corrected by _Vaclav Kotesovec_, Feb 17 2014
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