%I A159310
%S A159310 1,3,7,97,601,7576,116929,2482537,42814321,1040362966,25933795801,
%T A159310 760154969850,23297606120881,816970034324900,29137514248718373,
%U A159310 1194044411689941241,48661170952876980481,2227962859999303395766
%N A159310 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 A159310 G.f.: Sum_{n>=1} log(1 + a(n)*x^n/n!) = Sum_{n>=1} n^(n-1)*x^n/n! = -LambertW(-x).
%F A159310 G.f.: Sum_{n>=1} log(1 + a(n)*exp(-n*x)*x^n/n!) = x.
%F A159310 Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 15 2009: 
               (Start)
%F A159310 G.f.: Sum_{n>=1} n*a(n)*x^n/[n! + a(n)*x^n] = Sum_{n>=1} n^n*x^n/n!.
%F A159310 G.f.: Sum_{n>=1} n*a(n)*x^n/[n!*exp(nx) + a(n)*x^n] = x/(1-x).
%F A159310 Recurrence:
%F A159310 a(n) = n^(n-1) + (n-1)!*[(-1)^n + Sum_{d divides n, 1<d<n} d*( -a(d)/
               d! )^(n/d) ] for n>1 with a(1)=1.
%F A159310 (End)
%e A159310 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 A159310 W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! +...
%e A159310 where W(x/exp(x)) = exp(x) and exp(x*W(x)) = W(x) = LambertW(-x)/(-x).
%o A159310 (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 A159310 (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))))))} [From Paul D. Hanna (pauldhanna(AT)juno.com), 
               Apr 15 2009]
%Y A159310 Cf. A137852.
%Y A159310 Sequence in context: A088419 A062592 A074349 this_sequence A129660 A158467 
               A028414
%Y A159310 Adjacent sequences: A159307 A159308 A159309 this_sequence A159311 A159312 
               A159313
%K A159310 nonn
%O A159310 1,2
%A A159310 Paul D. Hanna (pauldhanna(AT)juno.com), Apr 15 2009