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%I #10 Jul 19 2016 11:17:56
%S 1,2,10,68,544,4832,46312,471536,5055328,56795840,667286656,
%T 8197599104,105446118784,1423627264256,20234885027968,303737480337152,
%U 4827671316780544,81385455480335360,1455806861755411456
%N G.f.: A(x) = Series_Reversion( x / Sum_{n>=0} (n+1)!*x^n ).
%F G.f. A(x) satisfies:
%F (1) A(x) = F(x*A(x)) and A(x/F(x)) = F(x);
%F (2) a(n) = [x^n] F(x)^n / (n+1);
%F where F(x) = Sum_{n>=0} (n+1)!*x^n.
%F G.f. satisfies: A(x) = 1 + 2*Sum_{n>=1} n^n * x^n * A(x)^n / (1 + n*x*A(x))^n. - _Paul D. Hanna_, Feb 04 2013
%e G.f.: A(x) = 1 + 2*x + 10*x^2 + 68*x^3 + 544*x^4 + 4832*x^5 + 46312*x^6 +...
%e Let F(x) = 1 + 2x + 6x^2 + 24x^3 + 120x^4 + 720x^5 +...+ (n+1)!*x^n +...
%e then A(x) = F(x*A(x)) and A(x/F(x)) = F(x);
%e also, a(n) = coefficient of x^n in F(x)^n divided by (n+1).
%e The g.f. A(x) also satisfies:
%e A(x) = 1 + 2*x*A(x)/(1+x*A(x)) + 2*2^2*x^2*A(x)^2/(1+2*x*A(x))^2 + 2*3^3*x^3*A(x)^3/(1+3*x*A(x))^3 + 2*4^4*x^4*A(x)^4/(1+4*x*A(x))^4 +...
%o (PARI) a(n)=polcoeff(serreverse(x/sum(k=0,n,(k+1)!*x^k +x*O(x^n))),n)
%o (PARI) a(n)=local(A=1+x); for(i=1, n, A=1+2*sum(m=1, n, m^m*x^m*A^m/(1+m*x*A+x*O(x^n))^m)); polcoeff(A, n)
%o for(n=0, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Feb 04 2013
%Y Cf. A090753, A075834, A211207.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 14 2008
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