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%I #12 Mar 11 2015 01:36:04
%S 1,1,4,32,392,6492,135876,3450048,103437240,3592136052,142420085352,
%T 6373515782544,318952925294496,17709716546754768,1083721599571837632,
%U 72661988008040494752,5310638869813310865984,421165309915181979699168,36095229917026934599948992
%N G.f.: A(x) = x + x*ITERATE^2(x + 2*x*ITERATE^2(x + 3*x*ITERATE^2(x + 4*x*ITERATE^2(x + ...)))), where ITERATE^2(F(x)) = F(F(x)), and the nested iterations continue indefinitely.
%H Paul D. Hanna, <a href="/A195193/b195193.txt">Table of n, a(n) for n = 1..100</a>
%e G.f.: A(x) = x + x^2 + 4*x^3 + 32*x^4 + 392*x^5 + 6492*x^6 +...
%e where A(x) is generated by nested iterations of shifted series:
%e A(x) = x + x*B(B(x)), where
%e B(x) = x + 2*x^2 + 12*x^3 + 132*x^4 + 2118*x^5 + 44400*x^6 +...;
%e B(x) = x + 2*x*C(C(x)), where
%e C(x) = x + 3*x^2 + 24*x^3 + 336*x^4 + 6672*x^5 + 169560*x^6 +...;
%e C(x) = x + 3*x*D(D(x)), where
%e D(x) = x + 4*x^2 + 40*x^3 + 680*x^4 + 16100*x^5 + 481200*x^6 +...;
%e D(x) = x + 4*x*E(E(x)), where
%e E(x) = x + 5*x^2 + 60*x^3 + 1200*x^4 + 33000*x^5 + 1134420*x^6 +...;
%e E(x) = x + 5*x*F(F(x)), where
%e F(x) = x + 6*x^2 + 84*x^3 + 1932*x^4 + 60522*x^5 + 2352672*x^6 +...;
%e E(x) = x + 6*x*G(G(x)), where
%e G(x) = x + 7*x^2 + 112*x^3 + 2912*x^4 + 102368*x^5 + 4440240*x^6 +...;
%e G(x) = x + 7*x*H(H(x)), where
%e H(x) = x + 8*x^2 + 144*x^3 + 4176*x^4 + 162792*x^5 + 7794720*x^6 +...; ...
%e Also, the 2nd iteration of the g.f. A(x) begins:
%e A(A(x)) = x + 2*x^2 + 10*x^3 + 85*x^4 + 1044*x^5 + 16996*x^6 + 347960*x^7 + 8649376*x^8 + 254561796*x^9 + 8704440152*x^10 +...
%o (PARI) {a(n)=local(A=x); for(k=0, n, A=x + (n-k+1)*x*subst(A,x,A+x*O(x^n))); polcoeff(A, n)}
%K nonn
%O 1,3
%A _Paul D. Hanna_, Sep 11 2011
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