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

0,3

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

Formula

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

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

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 *)

Prog

(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

Crossrefs

Column k=2 of A287847.

Cf. A000108.

Sequence in context: A148190 A151434 A296292 * A148191 A141312 A148192

Adjacent sequences: A287963 A287964 A287965 * A287967 A287968 A287969

Keyword

nonn

Author

Alois P. Heinz, Jun 03 2017

Status

approved