A288116 Number of Dyck paths of semilength n such that each level has exactly nine peaks or no peaks.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 9, 37, 101, 227, 487, 1019, 2015, 3724, 6528, 11438, 24758, 81106, 330810, 1542486, 7723906, 35765450, 142808117, 494994177, 1533142713, 4370885515, 11737660709, 30111369545, 74286138919, 177289070957, 416431652499

Offset

0,12

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(j-1, i-1)+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, 9$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[j - 1, i - 1] + Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
a[n_] := If[n == 0, 1, b[n, 9, 9]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Crossrefs

Column k=9 of A288108.

Sequence in context: A299290 A304290 A244245 * A165394 A213570 A271908

Adjacent sequences: A288113 A288114 A288115 * A288117 A288118 A288119

Keyword

nonn

Author

Alois P. Heinz, Jun 05 2017

Status

approved