login
G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n * (Sum_{k=0..n} C(n,k)^2*x^k), where g.f. A(x) = Sum_{n>=1} a(n)*x^n.
1

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

%S 1,1,2,8,76,1452,45612,2095992,131601136,10790109464,1117867502280,

%T 142679360514256,21987281765799840,4023859534010994768,

%U 862536439626951197192,214034590216271750690880,60867125826968771742513120,19663683837171331703090010864

%N G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n * (Sum_{k=0..n} C(n,k)^2*x^k), 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 + 76*x^5 + 1452*x^6 + 45612*x^7 +...

%e where

%e A(A(x)) = x*(1+x) + x^2*(1+4*x+x^2) + 2*x^3*(1+9*x+9*x^2+x^3) + 8*x^4*(1+16*x+36*x^2+16*x^3+x^4) + 76*x^5*(1+25*x+100*x^2+100*x^3+25*x^4+x^5) +...

%e Explicitly,

%e A(A(x)) = x + 2*x^2 + 6*x^3 + 27*x^4 + 222*x^5 + 3642*x^6 + 105612*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*sum(k=0,m,binomial(m,k)^2*x^k) +x*O(x^#A));

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

%Y Cf. A193206.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jul 19 2011