A288319 Number of Dyck paths of semilength n such that each positive level has exactly three peaks.

1, 0, 0, 1, 0, 0, 0, 4, 20, 20, 0, 16, 200, 1120, 3540, 6864, 9400, 18240, 82000, 364256, 1255040, 3448400, 8094400, 18653984, 50789120, 166596240, 565558400, 1791310496, 5202559520, 14279014880, 39040502400, 111437733184, 335085082880, 1032287357600

Offset: 0

Author

Alois P. Heinz, Jun 07 2017

Keywords

• nonn

Examples

			. a(7) = 4:
.           /\/\/\        /\/\/\        /\/\/\        /\/\/\
.    /\/\/\/      \  /\/\/      \/\  /\/      \/\/\  /      \/\/\/\ .
		

Links

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

Crossrefs

Column k=3 of A288318.

Cf. A000108.

Programs

Maple

b:= proc(n, k, j) option remember;
     `if`(n=j, 1, add(b(n-j, k, i)*(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..35);

Mathematica

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(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, 35}] (* Jean-François Alcover, Jun 02 2018, from Maple *)