%I #9 Mar 30 2012 18:37:27

%S 1,1,2,8,63,866,18444,559083,22773527,1197061138,78782852673,

%T 6341384941543,612605031308910,69931195961966196,9310803519433216321,

%U 1429869869684956113511,250857267705012344767575,49858270430813771746874366,11143529422156562195864991584

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

%e G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 63*x^5 + 866*x^6 + 18444*x^7 +...

%e where

%e A(A(x)) = x/(1-x) + x^2/(1-x)^3 + 2*x^3/(1-x)^6 + 8*x^4/(1-x)^10 + 63*x^5/(1-x)^15 + 866*x^6/(1-x)^21 +...+ a(n)*x^n/(1-x)^(n*(n+1)/2) +...

%e Explicitly,

%e A(A(x)) = x + 2*x^2 + 6*x^3 + 27*x^4 + 196*x^5 + 2379*x^6 + 46224*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*(m+1)/2));

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

%Y Cf. A193192, A193193, A193194.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jul 19 2011