A288112 Number of Dyck paths of semilength n such that each level has exactly five peaks or no peaks.

1, 0, 0, 0, 0, 1, 1, 5, 11, 19, 32, 60, 178, 612, 1910, 6505, 22097, 62717, 155341, 365413, 908850, 2587326, 8337462, 28613490, 99865122, 341887279, 1112148217, 3385839203, 9723179369, 27116765041, 76656520298, 228493968174, 728697760582, 2447359432110

Offset

0,8

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, 5$2)):
seq(a(n), n=0..37);

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, 5, 5]];
Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Crossrefs

Column k=5 of A288108.

Sequence in context: A125003 A062718 A326123 * A225250 A105914 A289530

Adjacent sequences: A288109 A288110 A288111 * A288113 A288114 A288115

Keyword

nonn

Author

Alois P. Heinz, Jun 05 2017

Status

approved