A119471 - OEIS

%I #6 Nov 28 2023 11:12:42

%S 1,1,1,4,8,17,50,146,399,1087,3042,8741,25509,75259,223529,665215,

%T 1983226,5931158,17800505,53627756,162206221,492399027,1499501067,

%U 4579193127,14017819056,43001141630,132154209754,406818719006

%N G.f. A(x) equals the limit of the composition of functions (x+x^n) in reverse order; let F_1(x) = x, F_{n+1}(x) = F_n(x) + F_n(x)^(n+1), then A(x) = limit F_n(x): A(x) = ...o x+x^n o ... o x+x^3 o x+x^2 o x.

%H Paul D. Hanna, <a href="/A119471/b119471.txt">Table of n, a(n) for n = 1..500</a>

%e G.f.: A(x) is the limit of the composition of functions (x+x^n):

%e F_3(x) = x+x^3 o x+x^2 o x = x + x^2 + x^3 + 3*x^4 + 3*x^5 + x^6;

%e F_4(x) = x+x^4 o F_3(x) = x + x^2 + x^3 + 4*x^4 + 7*x^5 + 11*x^6 +...

%e F_5(x) = x+x^5 o F_4(x) = x + x^2 + x^3 + 4*x^4 + 8*x^5 + 16*x^6 +...

%e F_6(x) = x+x^6 o F_5(x) = x + x^2 + x^3 + 4*x^4 + 8*x^5 + 17*x^6 +...

%e F_7(x) = x+x^7 o x+x^6 o x+x^5 o x+x^4 o x+x^3 o x+x^2 o x =

%e x + x^2 + x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 50*x^7 + 145*x^8 +...

%o (PARI) {a(n)=local(F=x);if(n<1,0,for(k=2,n, F=subst(x+x^k,x,F+x*O(x^n)););return(polcoeff(F,n)))}

%Y Cf. A119470, A119472, A119459, A119460.

%K nonn

%O 1,4

%A _Paul D. Hanna_, May 22 2006