cp's OEIS Frontend

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

%I A112317 #13 Feb 11 2015 15:27:58
%S A112317 1,2,6,30,220,2170,27076,409836,7303164,149837028,3479498880,
%T A112317 90230486346,2584679465160,81056989408928,2762187020749144,
%U A112317 101633218030586364,4015771398425994048,169588657820702174728
%N A112317 Coefficients of x^n in the n-th iteration of (x + x^2) for n>=1.
%C A112317 Forms a diagonal of the tables A122888 and A185755.
%H A112317 Paul D. Hanna and Vaclav Kotesovec, <a href="/A112317/b112317.txt">Table of n, a(n) for n = 1..300</a> (first 100 terms from Paul D. Hanna)
%F A112317 a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(x+x^2) with F_1(x) = x+x^2.
%e A112317 The initial iterations of x + x^2 begin:
%e A112317 F(x) = (1)*x + x^2;
%e A112317 F(F(x)) = x + (2)*x^2 + 2*x^3 + x^4;
%e A112317 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;
%e A112317 F(F(F(F(x)))) = x + 4*x^2 + 12*x^3 + (30)*x^4 + 64*x^5 +...;
%e A112317 F(F(F(F(F(x))))) = x + 5*x^2 + 20*x^3 + 70*x^4 + (220)*x^5 +...;
%e A112317 F(F(F(F(F(F(x)))))) = x + 6*x^2 + 30*x^3 + 135*x^4 + 560*x^5 + (2170)*x^6 +...;
%e A112317 where the terms in parenthesis illustrate how to form this sequence.
%o A112317 (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)))}
%o A112317 for(n=1, 30, print1(a(n), ", "))
%Y A112317 Cf. A112319, A112320, A122888, A185755, A135080, A166900.
%K A112317 nonn
%O A112317 1,2
%A A112317 _Paul D. Hanna_, Sep 03 2005
%E A112317 Added cross-references and comments; name and example changed by _Paul D. Hanna_, Feb 04 2011