%I #15 Mar 25 2024 16:35:13
%S 0,0,0,0,1,4,20,80,315,1176,4284,15240,53295,183700,625768,2110472,
%T 7057505,23427600,77271120,253426752,827009523,2686728060,8693388060,
%U 28026897360,90058925649,288516259416,921755412900,2937377079000,9338728806225,29626186593276
%N Number of Dyck paths of semilength n having exactly three (possibly overlapping) occurrences of the consecutive step pattern UDU, where U = (1,1) and D = (1,-1).
%H Alois P. Heinz, <a href="/A371408/b371408.txt">Table of n, a(n) for n = 0..2090</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F a(n) mod 2 = A121262(n) for n >= 1.
%e a(4) = 1: UDUDUDUD.
%e a(5) = 4: UDUDUDUUDD, UDUDUUDUDD, UDUUDUDUDD, UUDUDUDUDD.
%p a:= n-> `if`(n<4, 0, binomial(n-1, 3)*add(binomial(n-3, j)*
%p binomial(n-3-j, j-1), j=0..ceil((n-3)/2))/(n-3)):
%p seq(a(n), n=0..29);
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n<5, [0$4, 1][n+1],
%p (n-1)*((2*n-7)*a(n-1)+3*(n-2)*a(n-2))/((n-2)*(n-4)))
%p end:
%p seq(a(n), n=0..29);
%Y Column k=3 of A091869.
%Y Cf. A000108, A001006, A005717, A102839, A121262.
%K nonn
%O 0,6
%A _Alois P. Heinz_, Mar 22 2024