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%I #12 Jul 19 2016 11:33:29
%S 1,2,12,124,1820,34476,796600,21647088,674894736,23727325600,
%T 929120132336,40145865303072,1899037891380448,97624128373666272,
%U 5416162739385169920,322369745413410839296,20496143199154076929280
%N E.g.f. satisfies A(A(x)*exp(-2*A(x)))=x
%F a(n)=n!*T(n,1), T(n,m)=-1/2*(sum(k=m+1..n-1, T(k,m)*sum(i=k..n, T(n,i)*(k^(i-k)*(-2)^(i-k))/(i-k)!))+sum(i=m+1..n, T(n,i)*(m^(i-m)*(-2)^(i-m))/(i-m)!)), T(n,n)=1.
%e E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 124*x^4/4! + 1820*x^5/5! +...
%e where
%e A(A(x)) = x + 4*x^2/2! + 36*x^3/3! + 512*x^4/4! + 10000*x^5/5! +...+ (2*n)^(n-1)*x^n/n! +...
%o (Maxima)
%o T(n,m):=if n=m then 1 else -1/2*(sum(T(k,m)*sum(T(n,i)*(k^(i-k)*(-2)^(i-k))/(i-k)!,i,k,n),k,m+1,n-1)+sum(T(n,i)*(m^(i-m)*(-2)^(i-m))/(i-m)!,i,m+1,n));
%o makelist(n!*T(n,1),n,1,7);
%o (PARI) {a(n)=local(W=sum(m=1,n,(2*m)^(m-1)*x^m/m!)+x*O(x^n),A=x);
%o for(i=1,n,A=(A+subst(W,x,serreverse(A+x*O(x^n))))/2);n!*polcoeff(A,n)}
%o for(n=1,20,print1(a(n),", ")) \\ _Paul D. Hanna_, May 28 2013
%K nonn
%O 1,2
%A _Vladimir Kruchinin_, Mar 11 2012
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