A288324 Number of Dyck paths of semilength n such that each positive level has exactly eight peaks.

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 9, 315, 3465, 17325, 45045, 63063, 45045, 12870, 0, 81, 6075, 200340, 3835755, 48617415, 440531784, 3000152925, 15896972520, 67174514550, 230430986514, 649879542063, 1519950287430, 2963421671535, 4828750295985

Offset

0,18

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Wikipedia, Counting lattice paths

Maple

b:= proc(n, k, j) option remember;
     `if`(n=j, 1, add(b(n-j, k, i)*(binomial(i, k)
      *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
    end:
a:= n-> `if`(n=0, 1, b(n, 8$2)):
seq(a(n), n=0..40);

Mathematica

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
a[n_] := If[n == 0, 1, b[n, 8, 8]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Crossrefs

Column k=8 of A288318.

Cf. A000108.

Sequence in context: A160194 A217145 A266835 * A317634 A198401 A135609

Adjacent sequences: A288321 A288322 A288323 * A288325 A288326 A288327

Keyword

nonn

Author

Alois P. Heinz, Jun 07 2017

Status

approved