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

0,12

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

Crossrefs

Column k=5 of A288386.

Cf. A000108.

Sequence in context: A200504 A267641 A267669 * A124092 A036293 A133679

Adjacent sequences: A288678 A288679 A288680 * A288682 A288683 A288684

Keyword

nonn

Author

Alois P. Heinz, Jun 13 2017

Status

approved