OFFSET
1,3
FORMULA
G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n * (Sum_{k=0..n-1} C(n-1,k)^2*x^k)/(1-x)^(2*n-1).
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 21*x^4 + 293*x^5 + 7025*x^6 +...
where
A(A(x)) = x/(1-x) + x^2*(1+x)/(1-x)^3 + 3*x^3*(1+4*x+x^2)/(1-x)^5 + 21*x^4*(1+9*x+9*x^2+x^3)/(1-x)^7 + 293*x^5*(1+16*x+36*x^2+16*x^3+x^4)/(1-x)^9 + 7025*x^6*(1+25*x+100*x^2+100*x^3+25*x^4+x^5)/(1-x)^11 +...
Explicitly,
A(A(x)) = x + 2*x^2 + 8*x^3 + 58*x^4 + 754*x^5 + 16776*x^6 + 585910*x^7 +...
PROG
(PARI) {a(n)=local(A=[1], F=x, G=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A);
G=sum(m=1, #A-1, A[m]*x^m*sum(k=0, n, binomial(m+k-1, k)^2*x^k) +x*O(x^#A));
A[#A]=Vec(G)[#A]-Vec(subst(F, x, F))[#A]); if(n<1, 0, A[n])}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 19 2011
STATUS
approved