A112317 Coefficients of x^n in the n-th iteration of (x + x^2) for n>=1.

1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028, 3479498880, 90230486346, 2584679465160, 81056989408928, 2762187020749144, 101633218030586364, 4015771398425994048, 169588657820702174728

Offset: 1

Author

Paul D. Hanna, Sep 03 2005

Keywords

• nonn

Comments

Forms a diagonal of the tables A122888 and A185755.

Examples

			The initial iterations of x + x^2 begin:
F(x) = (1)*x + x^2;
F(F(x)) = 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+ 8*x^6+ 4*x^7+ x^8;
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 + (2170)*x^6 +...;
where the terms in parenthesis illustrate how to form this sequence.
		

Links

• Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 1..300 (first 100 terms from Paul D. Hanna)

Crossrefs

Cf. A112319, A112320, A122888, A185755, A135080, A166900.

Programs

PARI

{a(n)=local(F=x+x^2, G=x+x*O(x^n));if(n<1,0, for(i=1,n,G=subst(F,x,G));return(polcoeff(G,n,x)))}
for(n=1, 30, print1(a(n), ", "))

Formula

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

Extensions

Added cross-references and comments; name and example changed by Paul D. Hanna, Feb 04 2011