A288681 Number of Dyck paths of semilength n such that no positive level has fewer than five peaks.

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 7, 98, 575, 2009, 5468, 13365, 30910, 70156, 170830, 531334, 2203895, 10091063, 44034478, 176213307, 650957418, 2258314543, 7491190627, 24204620623, 77794583961, 254583038843, 865776314524, 3087754003802, 11479621448305

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=5 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, 5, j], {j, 5, n}]]; Table[a[n], {n, 0, 35}] (* Indranil Ghosh, Aug 10 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, 5, j) for j in range(5, n + 1)])
print([a(n) for n in range(36)]) # Indranil Ghosh, Aug 10 2017