A308815 a(n) = 16^n*Sum_{i=0..n} 2^i*binomial(n, i)*Catalan(2*n-i).

1, 0, 512, 32768, 3014656, 285212672, 28521267200, 2950642532352, 313455303196672, 33990302461067264, 3747096045042008064, 418698249981751459840, 47315379509207945445376, 5398203160147467800936448, 620943817223351793348509696, 71934849808689842265157271552, 8385447350799836001482047488000

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

Table of n, a(n) for n=0..16.

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

Cf. A000108 (Catalan numbers).

Sequence in context: A254450 A254484 A253864 * A253536 A254383 A253977

Adjacent sequences: A308812 A308813 A308814 * A308816 A308817 A308818

Keyword

nonn

Author

Michel Marcus, Jun 26 2019

Status

approved