A288685 Number of Dyck paths of semilength n such that no positive level has fewer than nine peaks.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 462, 7217, 57783, 289400, 1043781, 3042593, 7833174, 18821247, 43417043, 97550980, 215243289, 469069428, 1020806036, 2342090587, 6886047798, 32238887181, 199504672863, 1232775909721, 6881782444707

Offset

0,20

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, 9, j], {j, 9, 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, 9, j) for j in range(9, n + 1)])
print([a(n) for n in range(41)]) # Indranil Ghosh, Aug 10 2017

Crossrefs

Column k=9 of A288386.

Cf. A000108.

Sequence in context: A354439 A180087 A233219 * A068235 A247598 A303207

Adjacent sequences: A288682 A288683 A288684 * A288686 A288687 A288688

Keyword

nonn

Author

Alois P. Heinz, Jun 13 2017

Status

approved