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%I #9 Mar 30 2012 18:37:23
%S 1,1,3,9,31,121,540,2692,14938,91599,613393,4443653,34574511,
%T 287092372,2529894464,23546940208,230556268991,2366688840376,
%U 25394369705598,284078044017592,3305577581826931,39928510807691616,499751895103778173
%N G.f.: A(x) = Product_{n>=1} 1/(1 - G_n(x)^n) where G(x) = x+x^2 and G_n(x) denotes the n-th iteration of G(x): G_n(x) = G_{n-1}(G(x)) with G_0(x)=x.
%F G.f.: A(x) = exp( Sum_{n>=1} Sum_{d|n} d*G_d(x)^n/n ) where G(x) = x+x^2 and G_n(x) denotes the n-th iteration of G(x).
%e G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 31*x^4 + 121*x^5 + 540*x^6 +...
%e Let G_n(x) denote the n-th iteration of G(x) = x + x^2, then
%e the logarithm of A(x) begins:
%e log(A(x)) = G(x) + [G(x)^2 + 2*G_2(x)^2]/2 + [G(x)^3 + 3*G_3(x)^3]/3 + [G(x)^4 + 2*G_2(x)^4 + 4*G_4(x)^4]/4 + [G(x)^5 + 5*G_5(x)^5]/5 +...
%e Explicitly,
%e log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 81*x^4/4 + 391*x^5/5 + 2159*x^6/6 + 13049*x^7/7 + 86257*x^8/8 + 618976*x^9/9 + 4763325*x^10/10 +...
%e The initial iterations of G(x) = x + x^2 begin:
%e G(G(x)) = x + 2*x^2 + 2*x^3 + x^4;
%e G_3(x) = x + 3*x^2 + 6*x^3 + 9*x^4 + 10*x^5 + 8*x^6 + 4*x^7 + x^8;
%e G_4(x) = x + 4*x^2 + 12*x^3 + 30*x^4 + 64*x^5 + 118*x^6 +...;
%e G_5(x) = x + 5*x^2 + 20*x^3 + 70*x^4 + 220*x^5 + 630*x^6 +...;
%e G_6(x) = x + 6*x^2 + 30*x^3 + 135*x^4 + 560*x^5 + 2170*x^6 +...;
%e See A122888 for a table of coefficients in iterations of x + x^2.
%e The g.f. equals the product:
%e A(x) = 1/[(1-x-x^2)*(1-(x+2*x^2+2*x^3+x^4)^2))*(1-(x+3*x^2+6*x^3+9*x^4+10*x^5+8*x^6+4*x^7+x^8)^3)...*(1-G_n(x)^n)*...]
%e where G_n(x) equals the n-th iteration of x+x^2.
%o (PARI) /* n-th Iteration of a function: */
%o {ITERATE(n,F,p)=local(G=x);for(i=1,n,G=subst(F,x,G+x*O(x^p)));G}
%o /* G.f.: */
%o {a(n)=local(F);F=exp(sum(m=1,n+1,sumdiv(m,d,d*ITERATE(d,x+x^2,n)^m/m)));polcoeff(F,n)}
%o (PARI) {a(n)=polcoeff(1/prod(k=1,n,1-ITERATE(k,x+x^2,n)^k),n)}
%Y Cf. A182970, A122888.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 18 2010
%E Name changed by _Paul D. Hanna_, Dec 19 2010
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