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G.f. A(x) equals an infinite symmetric composition of functions x+x^n, n=2,3,4,...
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%I #2 Mar 30 2012 18:37:17

%S 1,2,4,13,39,113,359,1161,3708,11961,38903,126885,415509,1365935,

%T 4501009,14856863,49115474,162590020,538828790,1787431873,5934532485,

%U 19718347615,65559507987,218095336290,725890070743,2417022095921

%N G.f. A(x) equals an infinite symmetric composition of functions x+x^n, n=2,3,4,...

%F G.f.: A(x) = x+x^2 o x+x^3 o...o x+x^n o...o (x) o...o x+x^n o...o x+x^3 o x+x^2.

%e G.f.: A(x) = x + 2*x^2 + 4*x^3 + 13*x^4 + 39*x^5 + 113*x^6 +...

%e G.f. A(x) equals the limit of symmetric compositions starting with:

%e (1) x+x^2 o x+x^2 = x + 2*x^2 + 2*x^3 + x^4 ;

%e (2) x+x^2 o x+x^3 o x+x^3 o x+x^2 = x + 2*x^2 + 4*x^3 + 11*x^4 +...;

%e (3) x+x^2 o x+x^3 o x+x^4 o x+x^4 o x+x^3 o x+x^2 = x + 2*x^2 + 4*x^3 + 13*x^4 +...

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

%Y Cf. A163136, A163135.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 16 2009