%I #9 Oct 14 2020 08:38:41

%S 1,2,10,74,710,8322,115018,1828962,32852526,657188258,14477811178,

%T 348100068698,9067809569750,254354791759298,7642986480897930,

%U 244923580410697938,8337728465913016926,300482221889444825154,11429089791630856291018,457542303069698601849194,19230862148761320966737254,846710680545018639230252418

%N G.f. A(x) satisfies: A(x) = x*(1 + 3*A(x)*A'(x)) / (1 + A(x)*A'(x)).

%C Compare to: C(x) = x*(1 + 2*C(x)*C'(x)) / (1 + C(x)*C'(x)) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

%H Paul D. Hanna, <a href="/A301932/b301932.txt">Table of n, a(n) for n = 1..400</a>

%F a(n) ~ c * 2^n * n!, where c = 0.181799839377767875340143846... - _Vaclav Kotesovec_, Oct 14 2020

%e G.f.: A(x) = x + 2*x^2 + 10*x^3 + 74*x^4 + 710*x^5 + 8322*x^6 + 115018*x^7 + 1828962*x^8 + 32852526*x^9 + 657188258*x^10 + ...

%e such that A = A(x) satisfies: A = x*(1 + 3*A*A')/(1 + A*A').

%e RELATED SERIES.

%e A(x)*A'(x) = x + 6*x^2 + 48*x^3 + 470*x^4 + 5448*x^5 + 73374*x^6 + 1132000*x^7 + 19752822*x^8 + 385285080*x^9 + 8311631702*x^10 + ...

%o (PARI) {a(n) = my(L=x); for(i=1,n, L = x*(1 + 3*L'*L)/(1 + L'*L +x*O(x^n)) ); polcoeff(L,n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A301930, A301931, A301933.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 28 2018