A288682 Number of Dyck paths of semilength n such that no positive level has fewer than six peaks.

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 8, 156, 1213, 5232, 16091, 41834, 100320, 229851, 513699, 1166304, 3068322, 11294356, 54431307, 271824026, 1253186445, 5233138157, 20031588131, 71538367677, 242280234545, 789260222205, 2507719402158, 7900354628357

Offset

0,14

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, 6, j], {j, 6, 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, 6, j) for j in range(6, n + 1)])
print([a(n) for n in range(36)]) # Indranil Ghosh, Aug 10 2017

Crossrefs

Column k=6 of A288386.

Cf. A000108.

Sequence in context: A302959 A188408 A089669 * A268543 A345317 A113668

Adjacent sequences: A288679 A288680 A288681 * A288683 A288684 A288685

Keyword

nonn

Author

Alois P. Heinz, Jun 13 2017

Status

approved