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A244236
Number of Dyck paths of semilength n having exactly one occurrence of the consecutive pattern UDUD.
3
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 23 2014
STATUS
approved