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A120056 Expansion of g.f. A(x) satisfying A(x)^2 = (1+x) * A(x*A(x)) with A(0) = 1.

%I A120056 #18 Jan 09 2024 08:50:34
%S A120056 1,1,1,3,14,91,737,7096,78766,987842,13797282,212285511,3567147508,
%T A120056 65001800879,1276876433466,26901212485406,605144725934525,
%U A120056 14477882030623891,367106501841287465,9834718739970491625
%N A120056 Expansion of g.f. A(x) satisfying A(x)^2 = (1+x) * A(x*A(x)) with A(0) = 1.
%C A120056 Note that if F(x)^2 = (1+x) * F(x*F(x)) with F(0) = 0, then F(x) is the g.f. of A367386. - _Paul D. Hanna_, Jan 08 2024
%H A120056 Vaclav Kotesovec, <a href="/A120056/b120056.txt">Table of n, a(n) for n = 0..200</a>
%F A120056 a(n) ~ c * n^n / (exp(n) * (log(2))^n), where c = 0.8876659274678... . - _Vaclav Kotesovec_, Aug 08 2014
%F A120056 From _Paul D. Hanna_, Jan 08 2024: (Start)
%F A120056 G.f. A(x) = Sum_{n>=0} a(n)*x^n and B(x) = x*A(x) and B(x) = x*A(x) satisfy the following formulas.
%F A120056 (1) A(x)^2 = (1+x) * A(x*A(x)).
%F A120056 (2) A(x) = (1+x)^(1/2) * (1 + B(x))^(1/4) * (1 + B(B(x)))^(1/8) * (1 + B(B(B(x))))^(1/16) * (1 + B(B(B(B(x)))))^(1/32) * ..., an infinite product involving iterations of B(x) = x*A(x).
%F A120056 (3.a) B(B(x)) = x*A(x)^3 / (1+x).
%F A120056 (3.b) B(B(B(x))) = x*A(x)^7 / ((1+x)^3 * (1 + x*A(x))).
%F A120056 (3.c) B(B(B(B(x)))) = x*A(x)^15 / ((1+x)^7 * (1 + x*A(x))^3 * (1 + x*A(x)^3/(1+x))).
%F A120056 (3.d) B(B(B(B(B(x))))) = x*A(x)^31 / ((1+x)^15 * (1+x*A(x))^7 * (1 + x*A(x)^3/(1+x))^3 * (1 + x*A(x)^7/((1+x)^3*(1+x*A(x))))).
%F A120056 The compositions of g.f. A(x) with the iterations of B(x) = x*A(x) begin
%F A120056 (4.a) A(B(x)) = A(x)^2 / (1+x).
%F A120056 (4.b) A(B(B(x))) = A(x)^4 / ((1+x)^2 * (1 + x*A(x))).
%F A120056 (4.c) A(B(B(B(x)))) = A(x)^8 / ((1+x)^4 * (1 + x*A(x))^2 * (1 + x*A(x)^3/(1+x))).
%F A120056 (4.d) A(B(B(B(B(x))))) = A(x)^16 / ((1+x)^8 * (1+x*A(x))^4 * (1 + x*A(x)^3/(1+x))^2 * (1 + x*A(x)^7/((1+x)^3*(1+x*A(x))))).
%F A120056 (End)
%e A120056 A(x) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 91*x^5 + 737*x^6 +...
%e A120056 A(x)^2 = 1 + 2*x + 3*x^2 + 8*x^3 + 35*x^4 + 216*x^5 + 1693*x^6 +...
%e A120056 A(x*A(x)) = 1 + x + 2*x^2 + 6*x^3 + 29*x^4 + 187*x^5 + 1506*x^6 +...
%o A120056 (PARI) {a(n)=local(A); if(n<1, n==0, A=1+x; for(i=1,n, A=subst(A,x,x*A+x*O(x^n))*(1+x)/A); polcoeff(A, n))}
%Y A120056 Cf. A088792, A367386.
%K A120056 nonn
%O A120056 0,4
%A A120056 _Paul D. Hanna_, Jun 05 2006