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A322736 G.f. satisfies: A(x) = Sum_{n>=0} 2^n * ( (1+x)^n - A(x)^(1/2) )^n / ( 3 - 2*(1+x)^n * A(x)^(1/2) )^(n+1).

%I A322736 #9 Jan 25 2019 12:21:01
%S A322736 1,2,8,96,2956,114992,5244896,277303392,16680895688,1124043943848,
%T A322736 83860544099264,6863636560150656,611673708807594944,
%U A322736 58982083391411043456,6120766911879901270784,680339106407429897733760,80661483112436517009089168,10162784535291704640507410016,1356175692780348173552997926272,191103836643650458447321745220736,28358934286111202643351952170366400,4420810085328675478052952299755080000
%N A322736 G.f. satisfies: A(x) = Sum_{n>=0} 2^n * ( (1+x)^n - A(x)^(1/2) )^n / ( 3 - 2*(1+x)^n * A(x)^(1/2) )^(n+1).
%F A322736 G.f. A(x) and B(x) = A(x)^(1/2) satisfy:
%F A322736 (1) A(x) = Sum_{n>=0} 2^n * ( (1+x)^n - B(x) )^n  / ( 3 - 2*(1+x)^n * B(x) )^(n+1),
%F A322736 (2) A(x) = Sum_{n>=0} 2^n * ( (1+x)^n + B(x) )^n  / ( 3 + 2*(1+x)^n * B(x) )^(n+1).
%e A322736 G.f.: A(x) = 1 + 2*x + 8*x^2 + 96*x^3 + 2956*x^4 + 114992*x^5 + 5244896*x^6 + 277303392*x^7 + 16680895688*x^8 + 1124043943848*x^9 + ...
%e A322736 such that A(x) and B = A(x)^(1/2) satisfy
%e A322736 A(x) = 1/(3 - 2*B)  +  2*((1+x) - B)/(3 - 2*(1+x)*B)^2  +  2^2*((1+x)^2 - B)^2/(3 - 2*(1+x)^2*B)^3  +  2^3*((1+x)^3 - B)^3/(3 - 2*(1+x)^3*B)^4  +  2^4*((1+x)^4 - B)^4/(3 - 2*(1+x)^4*B)^5  +  2^5*((1+x)^5 - B)^5/(3 - 2*(1+x)^5*B)^6 + ...
%e A322736 also,
%e A322736 A(x) = 1/(3 + 2*B)  +  2*((1+x) + B)/(3 + 2*(1+x)*B)^2  +  2^2*((1+x)^2 + B)^2/(3 + 2*(1+x)^2*B)^3  +  2^3*((1+x)^3 + B)^3/(3 + 2*(1+x)^3*B)^4  +  2^4*((1+x)^4 + B)^4/(3 + 2*(1+x)^4*B)^5  +  2^5*((1+x)^5 + B)^5/(3 + 2*(1+x)^5*B)^6 + ...
%o A322736 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( (1+x)^m - Ser(A)^(1/2) )^m *2^m / (3 - 2*(1+x)^m*Ser(A)^(1/2) )^(m+1) ) ) ); A[n+1]}
%o A322736 for(n=0, 30, print1(a(n), ", "))
%Y A322736 Cf. A317350, A322735.
%K A322736 nonn
%O A322736 0,2
%A A322736 _Paul D. Hanna_, Jan 25 2019