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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Wikipedia, Counting lattice paths

Example

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

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 *)

Crossrefs

Row sums of A288318.

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

Sequence in context: A019146 A102537 A131202 * A067955 A182509 A049965

Adjacent sequences: A287984 A287985 A287986 * A287988 A287989 A287990

Keyword

nonn

Author

Alois P. Heinz, Jun 03 2017

Status

approved