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%I #47 Apr 03 2024 05:02:44
%S 2,7,25,91,336,1254,4719,17875,68068,260338,999362,3848222,14858000,
%T 57500460,222981435,866262915,3370764540,13135064250,51250632510,
%U 200205672810,782920544640,3064665881940,12007086477750,47081501377326,184753963255176,725510446350004
%N a(n) = (1/2)*(binomial(2n, n) - binomial(2n-2, n-1)).
%C 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
%H Stefano Spezia, <a href="/A024482/b024482.txt">Table of n, a(n) for n = 2..1600</a>
%H Toufik Mansour and I. L. Ramirez, <a href="https://ajc.maths.uq.edu.au/pdf/81/ajc_v81_p447.pdf">Enumerations of polyominoes determined by Fuss-Catalan words</a>, Australas. J. Combin. 81 (3) (2021) 447-457, table 2.
%F a(n) = A051924(n)/2. - _Zerinvary Lajos_, Jan 16 2007
%F From _R. J. Mathar_, Nov 09 2018: (Start)
%F D-finite with recurrence n*a(n) - (5*n-4)*a(n-1) + 2*(2*n-5)*a(n-2) = 0.
%F n*(3*n-5)*a(n) - 2*(3*n-2)*(2*n-3)*a(n-1) = 0. (End)
%F a(n) ~ 3*2^(2*n-3)/sqrt(n*Pi). - _Stefano Spezia_, May 09 2023
%F From _G. C. Greubel_, Apr 03 2024: (Start)
%F a(n) = (3*n-2)*A000108(n-1)/2.
%F G.f.: ((1-x)*sqrt(1-4*x) - (1+x)*(1-4*x))/(2*(1-4*x)).
%F E.g.f.: (1/2)*( -1 - x + exp(2*x)*( (1-x)*BesselI(0, 2*x) + x*BesselI(1, 2*x) ) ). (End)
%e The path udUududD has two sawtooth patterns, shown in lower case.
%p 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
%t Table[(Binomial[2n,n]-Binomial[2n-2,n-1])/2,{n,2,30}] (* _Harvey P. Dale_, Mar 04 2011 *)
%o (Magma) [(3*n-2)*Catalan(n-1)/2: n in [2..40]]; // _G. C. Greubel_, Apr 03 2024
%o (SageMath) [(3*n-2)*catalan_number(n-1)/2 for n in range(2,41)] # _G. C. Greubel_, Apr 03 2024
%Y Cf. A000108, A097613, A225015.
%K nonn
%O 2,1
%A _Clark Kimberling_
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