A287901 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has at least one peak.

1, 1, 1, 3, 6, 17, 49, 147, 459, 1476, 4856, 16282, 55466, 191474, 668510, 2356944, 8380944, 30025814, 108289093, 392871484, 1432934360, 5251507624, 19329771911, 71430479820, 264914270527, 985737417231, 3679051573264, 13769781928768, 51670641652576

Offset: 0

Author

Alois P. Heinz, Jun 02 2017

Keywords

• nonn

Examples

			. a(3) = 3:
.                   /\      /\
.      /\/\/\    /\/  \    /  \/\ .
.
. a(4) = 6:
.                    /\      /\        /\/\    /\        /\/\
.     /\/\/\/\  /\/\/  \  /\/  \/\  /\/    \  /  \/\/\  /    \/\ .
		

Links

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

Crossrefs

Column k=1 of A288386.

Cf. A000108, A281874, A287846.

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, 1, j], {j, n}]];Table[a[n], {n, 0, 30}] (* 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, 1, j) for j in range(1, n + 1)])
print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 09 2017