A288111 Number of Dyck paths of semilength n such that each level has exactly four peaks or no peaks.

1, 0, 0, 0, 1, 1, 4, 7, 11, 22, 81, 235, 673, 2063, 5716, 13627, 33752, 95729, 298232, 946563, 2977953, 9147328, 27004159, 76880498, 217826819, 637089405, 1949908577, 6160707450, 19627448025, 61909478550, 191681762379, 583025396879, 1756696160636

Offset: 0

Author

Alois P. Heinz, Jun 05 2017

Keywords

• nonn

Examples

			. a(6) = 4:
.                                             /\/\/\/\
.  /\  /\/\/\    /\/\  /\/\    /\/\/\  /\    /        \
. /  \/      \  /    \/    \  /      \/  \  /          \  .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Counting lattice paths

Crossrefs

Column k=4 of A288108.

Programs

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, 4$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, 4, 4]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 02 2018, from Maple *)