A287987 Number of Dyck paths of semilength n such that all positive levels have the same number of peaks.

1, 1, 1, 3, 1, 8, 13, 13, 54, 132, 280, 547, 1219, 3904, 11107, 25082, 53777, 137751, 419831, 1257599, 3453557, 8911341, 22636845, 59890162, 172264224, 529706648, 1630328686, 4765347773, 13125989799, 35253234315, 97531470556, 287880507391, 894915519516

Offset: 0

Author

Alois P. Heinz, Jun 03 2017

Keywords

• nonn

Examples

			. a(3) = 3:                         /\        /\
.                    /\/\/\      /\/  \      /  \/\  .
.
. a(5) = 8:
.                       /\/\      /\/\      /\/\
.      /\/\/\/\/\  /\/\/    \  /\/    \/\  /    \/\/\
.
.            /\        /\          /\        /\
.         /\/  \      /  \/\    /\/  \      /  \/\
.      /\/      \  /\/      \  /      \/\  /      \/\  .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• Wikipedia, Counting lattice paths

Crossrefs

Row sums of A288318.

Cf. A000108, A287845, A287846, A287993, A288109.

Programs

Maple

b:= proc(n, k, j) option remember; `if`(n=j, 1,
       add(binomial(i, k)*binomial(j-1, i-1-k)
         *b(n-j, k, i), i=1+k..min(j+k, n-j)))
    end:
a:= n-> 1+add(b(n, j$2), j=1..n/2):
seq(a(n), n=0..33);

Mathematica

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Binomial[i, k]*Binomial[j - 1, i - 1 - k]*b[n - j, k, i], {i, 1 + k, Min[j + k, n - j]}]];
a[n_] := 1 + Sum[b[n, j, j], {j, 1, n/2}];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, May 24 2018, translated from Maple *)