%I #7 Mar 30 2012 18:37:27

%S 1,1,3,22,334,8831,359836,20845201,1625007715,163854289212,

%T 20739421240200,3218400384155498,600776969761195428,

%U 132793055529329858607,34298178516935957467888,10235014757932193318825335

%N G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n / (1-x)^(n^2), where g.f. A(x) = Sum_{n>=1} a(n)*x^n.

%e G.f.: A(x) = x + x^2 + 3*x^3 + 22*x^4 + 334*x^5 + 8831*x^6 +...

%e where

%e A(A(x)) = x/(1-x) + x^2/(1-x)^4 + 3*x^3/(1-x)^9 + 22*x^4/(1-x)^16 + 334*x^5/(1-x)^25 + 8831*x^6/(1-x)^36 +...+ a(n)*x^n/(1-x)^(n^2) +...

%e Explicitly,

%e A(A(x)) = x + 2*x^2 + 8*x^3 + 60*x^4 + 842*x^5 + 20704*x^6 + 805796*x^7 +...

%o (PARI) {a(n)=local(A=[1],F=x,G=x);for(i=1,n,A=concat(A,0);F=x*Ser(A);

%o G=sum(m=1,#A-1,A[m]*x^m/(1-x+x*O(x^#A))^(m^2));

%o A[#A]=Vec(G)[#A]-Vec(subst(F,x,F))[#A]);if(n<1,0,A[n])}

%Y Cf. A193192, A193194, A193195.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jul 19 2011