A288679 Number of Dyck paths of semilength n such that no positive level has fewer than three peaks.

1, 0, 0, 1, 1, 1, 1, 5, 30, 96, 245, 592, 1543, 4884, 17660, 64495, 226442, 766937, 2558655, 8590293, 29408344, 102893203, 366035420, 1314955687, 4747101946, 17184305311, 62359953380, 226978626707, 829122987011, 3040369502702, 11191473790567, 41342469523031

Offset: 0

Author

Alois P. Heinz, Jun 13 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Wikipedia, Counting lattice paths

Crossrefs

Column k=3 of A288386.

Cf. A000108.

Programs

Mathematica

b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[k, i - j], i - 1}] b[n - j, k, i], {i, n - j}]];  a[n_]:=If[n==0, 1, Sum[b[n, 3, j], {j, 3, n}]];Table[a[n], {n, 0, 35}] (* Indranil Ghosh, Aug 09 2017 *)

Python

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum([sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(k, i - j), i)])*b(n - j, k, i) for i in range(1, n - j + 1)])
def a(n): return 1 if n==0 else sum([b(n, 3, j) for j in range(3, n + 1)])
print([a(n) for n in range(36)]) # Indranil Ghosh, Aug 09 2017