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A199548 G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 - A(x)^(2*n+1))/(1 - A(x)).

%I A199548 #12 Feb 10 2024 13:57:44
%S A199548 1,3,14,88,650,5257,45017,401010,3677344,34481492,329082191,
%T A199548 3186043296,31214870385,308901931412,3083146893716,31001118379636,
%U A199548 313734072027372,3193097704841990,32662597147529218,335616736745247652,3462524444288857191,35853293611333010079
%N A199548 G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 - A(x)^(2*n+1))/(1 - A(x)).
%C A199548 Compare to g.f. B(x) of A007317 (binomial transform of Catalan numbers):
%C A199548 B(x) = Sum_{n>=0} x^n * (1 - B(x)^(n+1))/(1 - B(x)).
%F A199548 G.f. satisfies: A(x) = 1 + x*A(x)*(2 + A(x)^2) - x^2*A(x)^3.
%F A199548 G.f. satisfies: x = (2 + A(x)^2 - sqrt(4 + 4*A(x) + A(x)^4))/(2*A(x)^2).
%F A199548 D-finite with recurrence 2*n*(2*n+1)*a(n) +(-47*n^2+31*n-2)*a(n-1) +4*(2*n^2+53*n-88)*a(n-2) +(255*n^2-1693*n+2668)*a(n-3) +2*(-217*n^2+1622*n-3028)*a(n-4) +2*(139*n^2-1235*n+2746)*a(n-5) -32*(n-5)*(2*n-11)*a(n-6)=0. - _R. J. Mathar_, Feb 10 2024
%e A199548 G.f.: A(x) = 1 + 3*x + 14*x^2 + 88*x^3 + 650*x^4 + 5257*x^5 +...
%e A199548 where g.f. A = A(x) satisfies the equivalent expressions:
%e A199548 A = 1 + x*(1-A^3)/(1-A) + x^2*(1-A^5)/(1-A) + x^3*(1-A^7)/(1-A) +...
%e A199548 A = 1 + x*(1 + A + A^2) + x^2*(1 + A + A^2 + A^3 + A^4) + x^3*(1 + A + A^2 + A^3 + A^4 + A^5 + A^6) +...
%o A199548 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*sum(k=0,2*m,A^k)+x*O(x^n)));polcoeff(A,n)}
%o A199548 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A*(2+A^2)-x^2*A^3+x*O(x^n));polcoeff(A,n)}
%Y A199548 Cf. A007317.
%K A199548 nonn
%O A199548 0,2
%A A199548 _Paul D. Hanna_, Nov 07 2011