A288686 Number of Dyck paths of semilength n such that no positive level has fewer than ten peaks.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 618, 11695, 112434, 665219, 2756389, 8890492, 24410518, 60972735, 144203914, 329766287, 737405644, 1623087349, 3531560786, 7633789153, 16745585892, 41482511559, 152244106469, 886899776271

Offset

0,22

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

Crossrefs

Column k=10 of A288386.

Cf. A000108.

Sequence in context: A218999 A220368 A220312 * A013587 A126159 A220992

Adjacent sequences: A288683 A288684 A288685 * A288687 A288688 A288689

Keyword

nonn

Author

Alois P. Heinz, Jun 13 2017

Status

approved