A288110 Number of Dyck paths of semilength n such that each level has exactly three peaks or no peaks.

1, 0, 0, 1, 1, 3, 4, 10, 36, 83, 225, 573, 1444, 3996, 11840, 34057, 95573, 267643, 754744, 2167250, 6347944, 18754719, 55183269, 161366349, 471263668, 1382569548, 4085677052, 12145287569, 36193473369, 107824201547, 320874528844, 954819540526, 2845349212512

Offset

0,6

Links

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

Wikipedia, Counting lattice paths

Example

a(5) = 3:
.                                  /\/\/\
.         /\  /\/\    /\/\  /\    /      \
.        /  \/    \  /    \/  \  /        \  .

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, 3$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, 3, 3]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Crossrefs

Column k=3 of A288108.

Sequence in context: A274220 A299881 A218293 * A085386 A290517 A295824

Adjacent sequences: A288107 A288108 A288109 * A288111 A288112 A288113

Keyword

nonn

Author

Alois P. Heinz, Jun 05 2017

Status

approved