A243871 Number of Dyck paths of semilength n having exactly 1 occurrence of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

1, 3, 10, 35, 124, 454, 1684, 6305, 23781, 90209, 343809, 1315499, 5050144, 19442366, 75034354, 290203076, 1124511549, 4364693311, 16966567970, 66041815437, 257378634365, 1004167036295, 3921726323436, 15330264382726, 59977821022143, 234839855088313

Offset: 5

Author

Alois P. Heinz, Jun 13 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..1000

Crossrefs

Column k=1 of A243881.

Column k=738 of A243827.

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, 2)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 1):
seq(a(n), n=5..40);

Formula

a(n) = (2*(2*n-17) *(2*n-19) *(2*n-9) *a(n-1) -(2*n-19) *(6*n^2-75*n+208) *a(n-4) +2*(2*n-17) *(10*n^2-136*n+387) *a(n-5) -(2*n-19) *(6*n^2-75*n+212) *a(n-8) +(32*n^3-704*n^2+4940*n-10850) *a(n-9) -(2*n-17) *(2*n-9) *(n-14) *a(n-10) -(2*n-19) *(n-8) *(2*n-9) *a(n-12) +2*(2*n-9) *(2*n^2-36*n+161) *a(n-13) -(n-10) *(2*n-17) *(2*n-9) *a(n-14)) / ((2*n-17) *(2*n-19) *(n-4)).