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A222589 G.f. satisfies: A(x) = Sum_{n>=0} x^n*(1 + n*x)^n * A(x)^n / (1 + x*A(x) + n*x^2*A(x))^n.

%I A222589 #6 Jan 15 2025 19:45:40
%S A222589 1,1,2,5,14,41,128,409,1355,4564,15728,54904,194740,698042,2532483,
%T A222589 9270351,34268276,127677731,479723132,1815553953,6923744832,
%U A222589 26587139445,102838915279,400513959602,1571152132075,6206954038519,24705172805012,99071049959707,400475021255313
%N A222589 G.f. satisfies: A(x) = Sum_{n>=0} x^n*(1 + n*x)^n * A(x)^n / (1 + x*A(x) + n*x^2*A(x))^n.
%F A222589 G.f. satisfies: A(x) = 1/2 + (1 + 2*x*A(x))/2 * Sum_{n>=0} (n+1)! * x^(2*n) * A(x)^n.
%e A222589 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 41*x^5 + 128*x^6 + 409*x^7 +...
%e A222589 where:
%e A222589 A(x) = 1 + x*(1+x)*A(x)/(1+x*(1+x)*A(x)) + x^2*(1+2*x)^2*A(x)^2/(1+x*(1+2*x)*A(x))^2 + x^3*(1+3*x)^3*A(x)^3/(1+x*(1+3*x)*A(x))^3 + x^4*(1+4*x)^4*A(x)^4/(1+x*(1+4*x)*A(x))^4 +...
%e A222589 Also,
%e A222589 A(x) = 1/2 + (1 + 2*x*A(x))/2 * (1 + 2*x^2*A(x) + 6*x^4*A(x)^2 + 24*x^6*A(x)^3 + 120*x^8*A(x)^4 + 720*x^10*A(x)^5 + 5040*x^12*A(x)^6 +...).
%o A222589 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, (x+m*x^2)^m*A^m / (1 + x*A+m*x^2*A +x*O(x^n))^m));polcoeff(A, n)}
%o A222589 for(n=0, 25, print1(a(n), ", "))
%o A222589 (PARI)
%o A222589 {a(n)=local(A=1+x);for(i=1,n,A=1/2+(1+2*x*A)*sum(k=0,n,(k+1)!/2*x^(2*k)*(A+x*O(x^n))^k));polcoeff(A, n)}
%o A222589 for(n=0, 25, print1(a(n), ", "))
%Y A222589 Cf. A187741.
%K A222589 nonn
%O A222589 0,3
%A A222589 _Paul D. Hanna_, Feb 25 2013