A243872 Number of Dyck paths of semilength n having exactly 2 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

1, 4, 16, 65, 263, 1077, 4419, 18132, 74368, 304778, 1247972, 5105477, 20867862, 85219608, 347724794, 1417697157, 5775652743, 23512922998, 95657223246, 388912046916, 1580241458120, 6417249216667, 26046042351889, 105661066012240, 428430870576913

Offset: 9

Author

Alois P. Heinz, Jun 13 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..400
• Vaclav Kotesovec, Recurrence (of order 14)

Crossrefs

Column k=2 of A243881.

Column k=738 of A243828.

Programs

Maple

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
     series(b(x-1, y+1, [2, 2, 4, 5, 6, 2, 4, 2, 10, 2][t])+`if`(t=10,
      z, 1)*b(x-1, y-1, [1, 3, 1, 3, 3, 7, 8, 9, 1, 3][t]), z, 3)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 2):
seq(a(n), n=9..40);

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, 1, Series[
      b[x-1, y+1, {2, 2, 4, 5, 6, 2, 4, 2, 10, 2}[[t]]]+If[t==10, z, 1]*
      b[x-1, y-1, {1, 3, 1, 3, 3, 7, 8, 9, 1, 3}[[t]]], {z, 0, 3}]]];
a[n_] := Coefficient[b[2n, 0, 1], z, 2];
a /@ Range[9, 40] (* Jean-François Alcover, Dec 27 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * sqrt(n), where d = 3.992152919721564592666177480042427843835641823811... is the root of equation 1 - 2*d + d^2 - 6*d^5 + 2*d^6 - 4*d^9 + d^10 = 0, and c = 0.00000109315704269290466088403991068... . - Vaclav Kotesovec, Jul 16 2014