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

0,5

Links

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

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

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 *)

Crossrefs

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

Sequence in context: A126729 A336006 A098730 * A163608 A081496 A152051

Adjacent sequences: A244233 A244234 A244235 * A244237 A244238 A244239

Keyword

nonn

Author

Alois P. Heinz, Jun 23 2014

Status

approved