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A371411 Number of Dyck paths of semilength 2n having exactly n (possibly overlapping) occurrences of the consecutive step pattern UDU, where U = (1,1) and D = (1,-1).

%I A371411 #14 Mar 26 2024 16:14:37
%S A371411 1,1,3,20,140,1134,9702,87516,817245,7852130,77135630,771742608,
%T A371411 7839348244,80661853300,839138980500,8813312133840,93339369441540,
%U A371411 995827949882370,10694044148599350,115515073043785800,1254354063204682440,13685749828961247180
%N A371411 Number of Dyck paths of semilength 2n having exactly n (possibly overlapping) occurrences of the consecutive step pattern UDU, where U = (1,1) and D = (1,-1).
%H A371411 Alois P. Heinz, <a href="/A371411/b371411.txt">Table of n, a(n) for n = 0..932</a>
%H A371411 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A371411 a(n) = A091869(2n,n).
%F A371411 a(n) mod 2 = 1 <=> n in { round(2^(2*k-3)) : k >= 0 } = { A081294 } U { 0 }.
%e A371411 a(1) = 1: UDUD.
%e A371411 a(2) = 3: UDUDUUDD, UDUUDUDD, UUDUDUDD.
%e A371411 a(3) = 20: UDUDUDUUDDUD, UDUDUDUUUDDD, UDUDUUDDUDUD, UDUDUUDUDDUD, UDUDUUDUUDDD, UDUDUUUDUDDD, UDUUDDUDUDUD, UDUUDUDDUDUD, UDUUDUDUDDUD, UDUUDUDUUDDD, UDUUDUUDUDDD, UDUUUDUDUDDD, UUDDUDUDUDUD, UUDUDDUDUDUD, UUDUDUDDUDUD, UUDUDUDUDDUD, UUDUDUDUUDDD, UUDUDUUDUDDD, UUDUUDUDUDDD, UUUDUDUDUDDD.
%e A371411 a(4) = 140: UDUDUDUDUUDDUUDD, UDUDUDUDUUUDDDUD, UDUDUDUDUUUDDUDD, ..., UUUDUDUUDUDUDDDD, UUUDUUDUDUDUDDDD, UUUUDUDUDUDUDDDD.
%p A371411 a:= proc(n) option remember; `if`(n<2, 1, (2*(n-1)*(2*n-1)^2*
%p A371411       a(n-1)+12*(n-2)*(2*n-1)*(2*n-3)*a(n-2))/((n+1)*n*(n-1)))
%p A371411     end:
%p A371411 seq(a(n), n=0..21);
%Y A371411 Cf. A000108, A081294, A091869.
%K A371411 nonn
%O A371411 0,3
%A A371411 _Alois P. Heinz_, Mar 22 2024