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A143508 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^2)^2.

%I A143508 #19 Mar 01 2025 08:35:06
%S A143508 1,1,2,9,52,372,3058,28074,282028,3059328,35497672,437499541,
%T A143508 5696752234,78036803430,1120687989348,16823652188164,263345788211608,
%U A143508 4289062071449610,72543038644585822,1271980596430351862,23085579883157411532,433071407705851089244
%N A143508 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^2)^2.
%F A143508 Given g.f. A(x), let G(x) be defined by G(x*A(x)^2) = x, then
%F A143508 (1) G(x) = x/(1 + A(x)^2*G(x))^2,
%F A143508 (2) A(G(x)) = 1 + A(x)^2*G(x).
%F A143508 From _Seiichi Manyama_, Mar 01 2025: (Start)
%F A143508 Let a(n,k) = [x^n] A(x)^k.
%F A143508 a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n-2*j+k,j)/(2*n-2*j+k) * a(n-j,2*j). (End)
%e A143508 G.f. A(x) = 1 + x + 2*x^2 + 9*x^3 + 52*x^4 + 372*x^5 + 3058*x^6 +...
%e A143508 A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 126*x^4 + 884*x^5 + 7149*x^6 +...
%e A143508 A(x*A(x)^2) = 1 + x + 4*x^2 + 22*x^3 + 156*x^4 + 1285*x^5 + 11886*x^6 +...
%e A143508 A(x*A(x)^2)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 372*x^4 + 3058*x^5 +...
%e A143508 Define G(x) by G(x*A(x)^2) = x, then
%e A143508 G(x) = x - 2*x^2 + 3*x^3 - 12*x^4 + 17*x^5 - 198*x^6 - 345*x^7 +...
%e A143508 such that G(x) = x/(1 + A(x)^2*G(x))^2.
%o A143508 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^2,x,x*A^2));polcoeff(A,n)}
%o A143508 (PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*j+k)*a(n-j, 2*j))); \\ _Seiichi Manyama_, Mar 01 2025
%Y A143508 Cf. A212029, A143501, A381568.
%K A143508 nonn
%O A143508 0,3
%A A143508 _Paul D. Hanna_, Aug 21 2008