A024482 a(n) = (1/2)*(binomial(2n, n) - binomial(2n-2, n-1)).

2, 7, 25, 91, 336, 1254, 4719, 17875, 68068, 260338, 999362, 3848222, 14858000, 57500460, 222981435, 866262915, 3370764540, 13135064250, 51250632510, 200205672810, 782920544640, 3064665881940, 12007086477750, 47081501377326, 184753963255176, 725510446350004

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Comments

Apparently the number of sawtooth patterns in all Dyck paths of semilength n, ([0,1],2,7,25,...). A sawtooth pattern is of the form (UD)^k, k >= 1. More generally, the number of sawtooth patterns of length > t in all Dyck paths with semilength (n+t), t >= 0. - David Scambler, Apr 23 2013

Examples

			The path udUududD has two sawtooth patterns, shown in lower case.
		

Links

• Stefano Spezia, Table of n, a(n) for n = 2..1600
• Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 2.

Crossrefs

Cf. A000108, A097613, A225015.

Programs

Magma

[(3*n-2)*Catalan(n-1)/2: n in [2..40]]; // G. C. Greubel, Apr 03 2024

Maple

Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z)/2: Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=2..25); # Zerinvary Lajos, Jan 16 2007

Mathematica

Table[(Binomial[2n,n]-Binomial[2n-2,n-1])/2,{n,2,30}] (* Harvey P. Dale, Mar 04 2011 *)

SageMath

[(3*n-2)*catalan_number(n-1)/2 for n in range(2,41)] # G. C. Greubel, Apr 03 2024

Formula

a(n) = A051924(n)/2. - Zerinvary Lajos, Jan 16 2007

From R. J. Mathar, Nov 09 2018: (Start)

D-finite with recurrence n*a(n) - (5*n-4)*a(n-1) + 2*(2*n-5)*a(n-2) = 0.

n*(3*n-5)*a(n) - 2*(3*n-2)*(2*n-3)*a(n-1) = 0. (End)

a(n) ~ 3*2^(2*n-3)/sqrt(n*Pi). - Stefano Spezia, May 09 2023

From G. C. Greubel, Apr 03 2024: (Start)

a(n) = (3*n-2)*A000108(n-1)/2.

G.f.: ((1-x)*sqrt(1-4*x) - (1+x)*(1-4*x))/(2*(1-4*x)).

E.g.f.: (1/2)*( -1 - x + exp(2*x)*( (1-x)*BesselI(0, 2*x) + x*BesselI(1, 2*x) ) ). (End)