A319578 a(n) = (1/3)*(n+2)^2*(3*n+3)!/(n+2)!^3.

1, 10, 140, 2310, 42042, 816816, 16628040, 350574510, 7595781050, 168212023980, 3792416540640, 86787993910800, 2011383287449200, 47123837020238400, 1114478745528638160, 26575401262863040830, 638330716607984804250, 15431925043610580004500, 375239440534109892741000

Offset: 0

Author

Peter Luschny, Sep 30 2018

Keywords

• nonn
• changed

Comments

Number of Schröder paths of length 2n+1 having n peaks.

Crossrefs

Cf. A007004, A060693, A215287.

Programs

Magma

[(1/3)*(n+2)^2*Factorial(3*n+3)/Factorial(n+2)^3: n in [0..20]]; // Vincenzo Librandi, Oct 01 2018

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 *)

PARI

a(n) = (1/3)*(n+2)^2*(3*n+3)!/(n+2)!^3; \\ Michel Marcus, Oct 01 2018

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