A288319 Number of Dyck paths of semilength n such that each positive level has exactly three peaks.

1, 0, 0, 1, 0, 0, 0, 4, 20, 20, 0, 16, 200, 1120, 3540, 6864, 9400, 18240, 82000, 364256, 1255040, 3448400, 8094400, 18653984, 50789120, 166596240, 565558400, 1791310496, 5202559520, 14279014880, 39040502400, 111437733184, 335085082880, 1032287357600

Offset

0,8

Links

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

Wikipedia, Counting lattice paths

Example

. a(7) = 4:
.           /\/\/\        /\/\/\        /\/\/\        /\/\/\
.    /\/\/\/      \  /\/\/      \/\  /\/      \/\/\  /      \/\/\/\ .

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, 3$2)):
seq(a(n), n=0..35);

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, 3, 3]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Crossrefs

Column k=3 of A288318.

Cf. A000108.

Sequence in context: A130316 A131745 A261755 * A330317 A151727 A146568

Adjacent sequences: A288316 A288317 A288318 * A288320 A288321 A288322

Keyword

nonn

Author

Alois P. Heinz, Jun 07 2017

Status

approved