A288110 Number of Dyck paths of semilength n such that each level has exactly three peaks or no peaks.

1, 0, 0, 1, 1, 3, 4, 10, 36, 83, 225, 573, 1444, 3996, 11840, 34057, 95573, 267643, 754744, 2167250, 6347944, 18754719, 55183269, 161366349, 471263668, 1382569548, 4085677052, 12145287569, 36193473369, 107824201547, 320874528844, 954819540526, 2845349212512

Offset: 0

Author

Alois P. Heinz, Jun 05 2017

Keywords

• nonn

Examples

			a(5) = 3:
.                                  /\/\/\
.         /\  /\/\    /\/\  /\    /      \
.        /  \/    \  /    \/  \  /        \  .
		

Links

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

Crossrefs

Column k=3 of A288108.

Programs

Maple

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
      b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
    end:
a:= n-> `if`(n=0, 1, b(n, 3$2)):
seq(a(n), n=0..40);

Mathematica

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n-j, k, i]*(Binomial[j-1, i-1] + Binomial[i, k]*Binomial[j-1, i-1-k]), {i, 1, Min[j+k, n-j]}]];
a[n_] := If[n == 0, 1, b[n, 3, 3]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 02 2018, from Maple *)