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

0, 0, 1, 1, 5, 14, 46, 150, 495, 1651, 5539, 18692, 63356, 215556, 735717, 2517941, 8637881, 29693938, 102263818, 352762106, 1218634659, 4215351719, 14598518663, 50611799048, 175639493624, 610076726280, 2120837219465, 7378415912617, 25687819032237

Offset: 0

Author

Alois P. Heinz, Jun 23 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=1 of A094507 and column k=10 of A243827.

Programs

Maple

a:= proc(n) option remember; `if`(n<5, [0$2, 1$2, 5][n+1],
     ((n-2)*(2*n-7)^2*a(n-1) +(28*n^3-212*n^2+501*n-361)*a(n-2)
      +(28*n^3-208*n^2+481*n-344)*a(n-3) +(n-3)*(2*n-3)^2*a(n-4)
      -(n-4)*(2*n-3)*(2*n-5)*a(n-5)) / ((n-1)*(2*n-5)*(2*n-7)))
    end:
seq(a(n), n=0..30);

Mathematica

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]]]];
a[n_] := Coefficient[b[2 n, 0, 1], z, 1];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz in A094507 *)

Formula

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