A193206 G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n * (Sum_{k>=0} C(n+k-1,k)^2*x^k), where g.f. A(x) = Sum_{n>=1} a(n)*x^n.

1, 1, 3, 21, 293, 7025, 257973, 13401901, 932658707, 83605059701, 9373165053231, 1284252646176937, 211069744296165851, 40975307787528699929, 9274936424952726229667, 2420958940356091360967253, 721771327240539119577124369

Offset

1,3

Links

Table of n, a(n) for n=1..17.

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

Cf. A193205.

Sequence in context: A222035 A361214 A171201 * A055555 A208731 A158888

Adjacent sequences: A193203 A193204 A193205 * A193207 A193208 A193209

Keyword

nonn

Author

Paul D. Hanna, Jul 19 2011

Status

approved