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%I #6 Feb 04 2018 03:17:09
%S 1,3,7,97,601,7576,116929,2482537,42814321,1040362966,25933795801,
%T 760154969850,23297606120881,816970034324900,29137514248718373,
%U 1194044411689941241,48661170952876980481,2227962859999303395766
%N G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! = LambertW(-x)/(-x).
%F G.f.: Sum_{n>=1} log(1 + a(n)*x^n/n!) = Sum_{n>=1} n^(n-1)*x^n/n! = -LambertW(-x).
%F G.f.: Sum_{n>=1} log(1 + a(n)*exp(-n*x)*x^n/n!) = x.
%F From _Paul D. Hanna_, Apr 15 2009: (Start)
%F G.f.: Sum_{n>=1} n*a(n)*x^n/(n! + a(n)*x^n) = Sum_{n>=1} n^n*x^n/n!.
%F G.f.: Sum_{n>=1} n*a(n)*x^n/(n!*exp(nx) + a(n)*x^n) = x/(1-x).
%F Recurrence:
%F a(n) = n^(n-1) + (n-1)!*((-1)^n + Sum_{d|n, 1<d<n} d*( -a(d)/d! )^(n/d) ) for n > 1 with a(1)=1.
%F (End)
%e G.f.: W(x) = (1+x)*(1+3*x^2/2!)*(1+7*x^3/3!)*(1+97*x^4/4!)*(1+601*x^5/5!)* ...
%e W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + ...
%e where W(x/exp(x)) = exp(x) and exp(x*W(x)) = W(x) = LambertW(-x)/(-x).
%o (PARI) {a(n)=if(n<1, 0, polcoeff(sum(k=0,n,(k+1)^(k-1)*x^k/k!)/prod(k=1, n-1, 1+a(k)*x^k +x*O(x^n)), n))}
%o (PARI) {a(n)=if(n<1, 0, if(n==1, 1,n^(n-1) + (n-1)!*((-1)^n + sumdiv(n, d, if(d<n&d>1, d*(-a(d)/d!)^(n/d))))))} \\ _Paul D. Hanna_, Apr 15 2009
%Y Cf. A137852.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Apr 15 2009
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