A288109 Number of Dyck paths of semilength n such that all levels with peaks have exactly the same number of peaks.

1, 1, 2, 5, 9, 23, 56, 122, 323, 792, 2060, 5199, 13314, 35171, 94077, 249285, 662901, 1775244, 4806724, 13125887, 36107283, 99863241, 276784435, 768288783, 2143763275, 6037486060, 17171063218, 49187617277, 141512589597, 408293870713, 1181084207303

Offset: 0

Author

Alois P. Heinz, Jun 05 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Wikipedia, Counting lattice paths

Crossrefs

Row sums of A288108.

Cf. A000108, A287987.

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-> 1 + add(b(n, j$2), j=1..n-1):
seq(a(n), n=0..33);

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_] := 1 + Sum[b[n, j, j], {j, 1, n - 1}];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, May 31 2018, from Maple *)