A288684 Number of Dyck paths of semilength n such that no positive level has fewer than eight peaks.

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 335, 4241, 27915, 117971, 373845, 1002089, 2456082, 5725439, 12935530, 28622833, 62588817, 139046970, 353173119, 1305216091, 7035422989, 41539474198, 227550374938, 1115122502718, 4917988882292

Offset

0,18

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

Wikipedia, Counting lattice paths

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, 8, j], {j, 8, n}]]; Table[a[n], {n, 0, 40}] (* Indranil Ghosh, Aug 10 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([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, 8, j) for j in range(8, n + 1)])
print([a(n) for n in range(41)]) # Indranil Ghosh, Aug 10 2017

Crossrefs

Column k=8 of A288386.

Cf. A000108.

Sequence in context: A218996 A352798 A113082 * A046747 A338799 A006426

Adjacent sequences: A288681 A288682 A288683 * A288685 A288686 A288687

Keyword

nonn

Author

Alois P. Heinz, Jun 13 2017

Status

approved