A287966 Number of Dyck paths of semilength n such that no level has more than two peaks.

1, 1, 2, 4, 12, 31, 90, 264, 797, 2402, 7355, 22725, 70573, 220007, 688379, 2160568, 6798020, 21428295, 67644503, 213806475, 676499166, 2142338437, 6789119425, 21527297986, 68292751071, 216737768906, 688082702872, 2185085230180, 6940609839680, 22050162168754

Offset: 0

Author

Alois P. Heinz, Jun 03 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 19.
• Wikipedia, Counting lattice paths

Crossrefs

Column k=2 of A287847.

Cf. A000108.

Programs

Mathematica

b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j, k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; a[n_]:=If[n==0, 1, m=Min[n, 2]; Sum[b[n, m, j], {j, m}]]; Table[a[n], {n, 0, 50}] (* Indranil Ghosh, Aug 17 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(b(n - j, k, i)*sum(binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)) for i in range(1, min(j + k, n - j) + 1))
def a(n):
    if n==0: return 1
    m=min(n, 2)
    return sum(b(n, m , j) for j in range(1, m + 1))
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 17 2017

Formula

a(n) = A287847(n,2).

a(n) = A000108(n) for n <= 2.