A112320 Coefficient of x^n in the (n+1)-th iteration of (x + x^2) for n>=1.

1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, 10052947476, 261595087182, 7509722346204, 235808741944100, 8040824716606176, 295914258931377276, 11690732617035570008, 493527339623630078552

Offset: 1

Author

Paul D. Hanna, Sep 06 2005

Keywords

• nonn

Examples

			The first few iterations of (x+x^2) begin:
F(x) = x + x^2;
F(F(x)) = (1)*x + 2*x^2 + 2*x^3 + x^4;
F(F(F(x))) = x + (3)*x^2 + 6*x^3 + 9*x^4 + 10*x^5 +...;
F(F(F(F(x)))) = x + 4*x^2 + (12)*x^3 + 30*x^4 + 64*x^5 +...;
F(F(F(F(F(x))))) = x + 5*x^2 + 20*x^3 + (70)*x^4 + 220*x^5 +...;
F(F(F(F(F(F(x)))))) = x + 6*x^2 + 30*x^3 + 135*x^4 + (560)*x^5 +...;
coefficients enclosed in parenthesis form the initial terms of this sequence.
		

Crossrefs

Cf. A112317, A112319.

Programs

PARI

{a(n)=local(F=x+x^2, G=x+x*O(x^n));if(n<1,0, for(i=1,n+1,G=subst(F,x,G));return(polcoeff(G,n,x)))}
for(n=1,25,print1(a(n),", "))

Formula

a(n) = [x^n] F_{n+1}(x) where F_{n+1}(x) = F_n(x+x^2) with F_1(x) = x+x^2 and F_0(x)=x for n>=1.