A288114 Number of Dyck paths of semilength n such that each level has exactly seven peaks or no peaks.

1, 0, 0, 0, 0, 0, 0, 1, 1, 7, 22, 48, 93, 180, 331, 575, 1150, 3578, 13268, 50808, 217173, 881980, 3064454, 9169075, 24605669, 61450068, 147038896, 347902716, 860591396, 2411664484, 8038395295, 30845855094, 126173520602, 513951305502, 1996531969713

Offset

0,10

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, 7$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, 7, 7]];
Table[a[n], {n, 0, 40}](* Jean-François Alcover, Jun 02 2018, from Maple *)

Crossrefs

Column k=7 of A288108.

Sequence in context: A244243 A223833 A014073 * A369549 A129109 A224141

Adjacent sequences: A288111 A288112 A288113 * A288115 A288116 A288117

Keyword

nonn

Author

Alois P. Heinz, Jun 05 2017

Status

approved