OFFSET
0,2
COMMENTS
Number of Schröder paths of length 2n+1 having n peaks.
FORMULA
a(n) = (n+2)*(3*n+2)!/((n+2)!^2*n!).
a(n) = A060693(2n+1,n).
G.f.: (hypergeom([1/3, 2/3], [2], 27*x) - 1)/(3*x). - Stefano Spezia, Aug 25 2025
a(n) ~ 3^(3*n+5/2) / (2 * n^2 * exp(1/(3*n)) * Pi). - Amiram Eldar, Sep 23 2025
MAPLE
a := n -> (n+2)*(3*n+2)!/((n+2)!^2*n!): seq(a(n), n = 0..18);
MATHEMATICA
Table[(n+2) (3*n+2)! / ((n+2)!^2 n!), {n, 0, 30}] (* Vincenzo Librandi, Oct 01 2018 *)
PROG
(PARI) a(n) = (1/3)*(n+2)^2*(3*n+3)!/(n+2)!^3; \\ Michel Marcus, Oct 01 2018
(Magma) [(1/3)*(n+2)^2*Factorial(3*n+3)/Factorial(n+2)^3: n in [0..20]]; // Vincenzo Librandi, Oct 01 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Luschny, Sep 30 2018
STATUS
approved