OFFSET
0,3
COMMENTS
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.
LINKS
Gavril Farkas, Riccardo Moschetti, Juan Carlos Naranjo, Gian Pietro Pirola, Alternating Catalan numbers and curves with triple ramification, arXiv:1906.10406 [math.AG], 2019.
FORMULA
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).
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
a(n) ~ 2^(7*n) / (3^(3/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 19 2021
PROG
(PARI) C(n) = binomial(2*n, n)/(n+1);
a(n) = 16^n*sum(i=0, n, (-2)^i*binomial(n, i)*C(2*n-i));
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Jun 26 2019
STATUS
approved