%I #14 Aug 11 2021 12:37:36
%S 1,1,3,13,71,469,3711,35181,398791,5352149,83650687,1494274301,
%T 29988083447,666634964197,16233361360559,429237520044813,
%U 12237655701598503,374023408217062261,12195222470567359071,422440153967133458205,15490152522612488256855,599350023954941335582725,24401304036660493806643215
%N G.f. A(x) satisfies: A(x) = 1 + x * A(x) * (A(x) + 3*x*A'(x)) / (A(x) + x*A'(x)).
%C Compare to: C(x) = 1 + x*C(x) * (C(x) + 2*x*C'(x)) / (C(x) + x*C'(x)) holds when C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H Paul D. Hanna, <a href="/A302699/b302699.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = 1 + x*A(x) * (A(x) + 3*x*A'(x)) / (A(x) + x*A'(x)).
%F (2) A(x) = 1/(1 - x*A(x)/(1 - x*A(x)/(1 - 2*x*A(x)/(1 - 3*x*A(x)/(1 - 4*x*A(x)/(1 - 5*x*A(x)/(1 - ...)))))), a continued fraction.
%F (3) A(x) = Series_Reversion( x - x^2*F(x) ) where F(x) = Sum_{n>=0} (2*n)!/(n!*2^n)*x^n (g.f. of the odd double factorials A001147).
%F a(n) ~ 2^(n - 1/2) * n^(n-1) / exp(n - 1/2). - _Vaclav Kotesovec_, Aug 11 2021
%e G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 71*x^4 + 469*x^5 + 3711*x^6 + 35181*x^7 + 398791*x^8 + 5352149*x^9 + 83650687*x^10 + ...
%e RELATED SERIES.
%e A'(x)/A(x) = 1 + 5*x + 31*x^2 + 225*x^3 + 1891*x^4 + 18473*x^5 + 210939*x^6 + 2815137*x^7 + 43551715*x^8 + 770297385*x^9 + ...
%e A(x) + x*A'(x) = 1 + 2*x + 9*x^2 + 52*x^3 + 355*x^4 + 2814*x^5 + 25977*x^6 + 281448*x^7 + 3589119*x^8 + 53521490*x^9 + ...
%o (PARI) /* Differential equation: */
%o {a(n) = my(A=1); for(i=0, n, A = 1 + x*A*(A + 3*x*A')/(x*A +x^2*O(x^n))'); polcoeff(G=A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* Continued fraction: */
%o {a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, for(k=0, n, CF = 1/(1 - (n-k+1)*x*A*CF ) ); A=1/(1 - x*A*CF) ); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A001147, A112934.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 11 2018