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A244236 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive pattern UDUD.

%I A244236 #13 Dec 21 2020 07:16:10
%S A244236 0,0,1,1,5,14,46,150,495,1651,5539,18692,63356,215556,735717,2517941,
%T A244236 8637881,29693938,102263818,352762106,1218634659,4215351719,
%U A244236 14598518663,50611799048,175639493624,610076726280,2120837219465,7378415912617,25687819032237
%N A244236 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive pattern UDUD.
%H A244236 Alois P. Heinz, <a href="/A244236/b244236.txt">Table of n, a(n) for n = 0..1000</a>
%F A244236 a(n) ~ c * (1/2+sqrt(2)+sqrt(5+4*sqrt(2))/2)^n / sqrt(n), where c = 0.0543819313385500572292392822783525275532509057751364636784836521... . - _Vaclav Kotesovec_, Jul 16 2014
%p A244236 a:= proc(n) option remember; `if`(n<5, [0$2, 1$2, 5][n+1],
%p A244236      ((n-2)*(2*n-7)^2*a(n-1) +(28*n^3-212*n^2+501*n-361)*a(n-2)
%p A244236       +(28*n^3-208*n^2+481*n-344)*a(n-3) +(n-3)*(2*n-3)^2*a(n-4)
%p A244236       -(n-4)*(2*n-3)*(2*n-5)*a(n-5)) / ((n-1)*(2*n-5)*(2*n-7)))
%p A244236     end:
%p A244236 seq(a(n), n=0..30);
%t A244236 b[x_, y_, t_] := b[x, y, t] = If[y < 0 || y > x, 0, If[x == 0, 1, Expand[ b[x - 1, y + 1, {2, 2, 4, 2}[[t]]] + b[x - 1, y - 1, {1, 3, 1, 3}[[t]]]* If[t == 4, z, 1]]]];
%t A244236 a[n_] := Coefficient[b[2 n, 0, 1], z, 1];
%t A244236 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ in A094507 *)
%Y A244236 Column k=1 of A094507 and column k=10 of A243827.
%K A244236 nonn
%O A244236 0,5
%A A244236 _Alois P. Heinz_, Jun 23 2014