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A171991 G.f. satisfies: A(x) = x + A(A(x))^2 + A(A(x))^4.

%I A171991 #24 May 04 2025 15:10:53
%S A171991 1,1,4,25,190,1644,15636,159977,1736392,19804646,235758596,2914539808,
%T A171991 37275052828,491727695628,6675266957376,93073877729749,
%U A171991 1330851640325930,19490214361291636,292025342161401292,4472427699080578554,69958787931298226280,1116911386287712897260
%N A171991 G.f. satisfies: A(x) = x + A(A(x))^2 + A(A(x))^4.
%F A171991 G.f. satisfies: A( x - A(x)^2 - A(x)^4 ) = x.
%e A171991 G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1644*x^6 + 15636*x^7 +...
%e A171991 Related series begin:
%e A171991 A(A(x)) = x + 2*x^2 + 10*x^3 + 71*x^4 + 598*x^5 + 5634*x^6 + 57624*x^7 +...
%e A171991 A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 182*x^5 + 1580*x^6 + 15080*x^7 +...
%e A171991 A(A(x))^4 = x^4 + 8*x^5 + 64*x^6 + 556*x^7 + 5192*x^8 + 51536*x^9 +...
%e A171991 A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 58*x^5 + 446*x^6 + 3868*x^7 + 36705*x^8 +...
%e A171991 A(x)^4 = x^4 + 4*x^5 + 22*x^6 + 152*x^7 + 1205*x^8 + 10564*x^9 +...
%e A171991 where the series reversion of the g.f. A(x) begins:
%e A171991 x - A(x)^2 - A(x)^4 = x - x^2 - 2*x^3 - 10*x^4 - 62*x^5 - 468*x^6 - 4020*x^7 -...
%t A171991 terms = 23; A[_] = 0; Do[A[x_] = x + A[A[x]]^2 + A[A[x]]^4 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* _Stefano Spezia_, May 04 2025 *)
%o A171991 (PARI) {a(n)=local(A=x+x^2,B=x); for(i=1, n, B=subst(A,x,A+x*O(x^n));A=x+B^2+B^4); polcoeff(A, n)}
%o A171991 (PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=serreverse(x-A^2-A^4+x*O(x^n))); polcoeff(A, n)}
%o A171991 for(n=1,30,print1(a(n),", "))
%Y A171991 Cf. A190761, A171992, A213591.
%K A171991 nonn
%O A171991 1,3
%A A171991 _Paul D. Hanna_, Jun 27 2012