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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

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-> 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 *)

Crossrefs

Row sums of A288108.

Cf. A000108, A287987.

Sequence in context: A088356 A246350 A192477 * A047044 A109621 A075200

Adjacent sequences: A288106 A288107 A288108 * A288110 A288111 A288112

Keyword

nonn

Author

Alois P. Heinz, Jun 05 2017

Status

approved