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%I #17 Aug 08 2019 20:47:21
%S 1,1,3,15,99,781,7001,69253,742071,8506775,103411463,1324477033,
%T 17785238513,249432247233,3642471258187,55246757713367,
%U 868523130653947,14127076257342933,237386074703124457,4115341407421082869,73516094755096807279,1351801707136238290351
%N G.f. satisfies A(x) = x + x*A(A(A(x))).
%F G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+2)) for n>0 with F(x,0)=1. - _Paul D. Hanna_, Apr 16 2007
%F G.f.: A(x) = G(x)/[1 + G(G(x))] where G(x) = A(A(x)) = g.f. of A141117.
%F G.f.: A(x) = Series_Reversion[ x/(1 + A(A(x))) ].
%F G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=0..n} A_{2*k}(x), where A_n(x) denotes the n-th iteration of A(x) with A_0(x)=x. - _Paul D. Hanna_, Jul 21 2011
%e G.f.: A(x) = x + x^2 + 3*x^3 + 15*x^4 + 99*x^5 + 781*x^6 +...
%e From _Paul D. Hanna_, Apr 16 2007:
%e G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
%e A = 1 + xC;
%e B = A*(1 + xD);
%e C = B*(1 + xE);
%e D = C*(1 + xF);
%e E = D*(1 + xG) ; ...
%e The solution to the variables in the system of equations are
%e A=A(x), B=A(A(x)), C=A(A(A(x))), D=A(A(A(A(x)))), etc., where:
%e A(x) = x + x^2 + 3*x^3 + 15*x^4 + 99*x^5 + 781*x^6 + 7001*x^7 +...
%e A(A(x)) = x + 2*x^2 + 8*x^3 + 46*x^4 + 330*x^5 + 2756*x^6 +...
%e A(A(A(x))) = x + 3*x^2 + 15*x^3 + 99*x^4 + 781*x^5 + 7001*x^6 +...
%e A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 180*x^4 + 1564*x^5 +...
%e A(A(A(A(A(x))))) = x + 5*x^2 + 35*x^3 + 295*x^4 + 2815*x^5 +...
%e ALTERNATE GENERATING METHOD.
%e The g.f. A(x) equals the sum of products of even iterations of A(x):
%e A(x) = x + x*A_2(x) + x*A_2(x)*A_4(x) + x*A_2(x)*A_4(x)*A_6(x) + x*A_2(x)*A_4(x)*A_6(x)*A_8(x) +...+ Product_{k=0..n} A_{2*k}(x) +...
%e where A_n(x) = A_{n-1}(A(x)) is the n-th iteration of A(x) with A_0(x)=x.
%t Nest[x + x (# /. x -> # /. x -> #) &, O[x], 30][[3]] (* _Vladimir Reshetnikov_, Aug 08 2019 *)
%o (PARI) {a(n)=local(A);A=x+x^2;for(i=3,n, A=x+x*subst(A,x,subst(A,x,A))+x*O(x^n)); polcoeff(A,n,x)}
%o (PARI) /* Define the n-th iteration of F: */
%o {ITERATE(F,n,p)=local(G=x);for(i=1,n,G=subst(F,x,G+x*O(x^p)));G}
%o /* A(x) equals the sum of products of even iterations of A(x): */
%o {a(n)=local(A=x);for(i=1,n,A=sum(m=0,n-1,prod(k=0,m,ITERATE(A,2*k,n)+x*O(x^n))));polcoeff(A,n)}
%Y Cf. A002449, A030266, A087949, A088714, A088717, A120971.
%Y Cf. A141117.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jan 31 2004, Jun 04 2008
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