%I #17 Nov 19 2021 08:24:35
%S 1,0,512,32768,3014656,285212672,28521267200,2950642532352,
%T 313455303196672,33990302461067264,3747096045042008064,
%U 418698249981751459840,47315379509207945445376,5398203160147467800936448,620943817223351793348509696,71934849808689842265157271552,8385447350799836001482047488000
%N a(n) = 16^n*Sum_{i=0..n} 2^i*binomial(n, i)*Catalan(2*n-i).
%C For n >= 3, a(n) is the number of odd covers of degree 2n+1 of a general curve of genus n. See Farkas et al.
%H Gavril Farkas, Riccardo Moschetti, Juan Carlos Naranjo, Gian Pietro Pirola, <a href="https://arxiv.org/abs/1906.10406">Alternating Catalan numbers and curves with triple ramification</a>, arXiv:1906.10406 [math.AG], 2019.
%F G.f.: 2*t/(sqrt(1+64*t^2+16*t*sqrt(16*t^2+1)) + sqrt(1+64*t^2-16*t*sqrt(16*t^2+1))) (odd powers only).
%F Conjecture: D-finite with recurrence: n*(2*n+1)*a(n) -32*(9*n-4)*(n-1)*a(n-1) +1024*(3*n^2-23*n+29)*a(n-2) +65536*(n-2)*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Jan 27 2020
%F a(n) ~ 2^(7*n) / (3^(3/2) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Nov 19 2021
%o (PARI) C(n) = binomial(2*n, n)/(n+1);
%o a(n) = 16^n*sum(i=0, n, (-2)^i*binomial(n, i)*C(2*n-i));
%Y Cf. A000108 (Catalan numbers).
%K nonn
%O 0,3
%A _Michel Marcus_, Jun 26 2019