OFFSET
2,1
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
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.
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)
EXAMPLE
The path udUududD has two sawtooth patterns, shown in lower case.
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 *)
PROG
(Magma) [(3*n-2)*Catalan(n-1)/2: n in [2..40]]; // G. C. Greubel, Apr 03 2024
(SageMath) [(3*n-2)*catalan_number(n-1)/2 for n in range(2, 41)] # G. C. Greubel, Apr 03 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved