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A024482 a(n) = (1/2)*(binomial(2n, n) - binomial(2n-2, n-1)). 5

%I

%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

%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

%F a(n) = A051924(n)/2. - _Zerinvary Lajos_, Jan 16 2007

%F From _R. J. Mathar_, Nov 09 2018: (Start)

%F 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)

%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 *)

%Y Cf. A097613, A225015.

%K nonn,changed

%O 2,1

%A _Clark Kimberling_

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Last modified January 18 11:50 EST 2019. Contains 319271 sequences. (Running on oeis4.)