A288683 Number of Dyck paths of semilength n such that no positive level has fewer than seven peaks.

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 9, 234, 2350, 12567, 44971, 127475, 320491, 756677, 1720610, 3821223, 8436508, 19793620, 59810128, 268048977, 1458971589, 7720465569, 36927931597, 159094351283, 626621217546, 2296016964863, 7949275945740

Offset

0,16

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, 7, j], {j, 7, 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, 7, j) for j in range(7, n + 1)])
print([a(n) for n in range(36)]) # Indranil Ghosh, Aug 10 2017

Crossrefs

Column k=7 of A288386.

Cf. A000108.

Sequence in context: A165389 A153223 A276536 * A302914 A157569 A279492

Adjacent sequences: A288680 A288681 A288682 * A288684 A288685 A288686

Keyword

nonn

Author

Alois P. Heinz, Jun 13 2017

Status

approved