A287846 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has exactly one peak.

1, 1, 0, 2, 0, 4, 6, 8, 24, 52, 96, 212, 504, 1072, 2352, 5288, 11928, 26800, 60336, 136304, 308928, 701248, 1593120, 3622016, 8245008, 18787360, 42836928, 97724384, 223052784, 509338816, 1163512032, 2658731648, 6077117376, 13893874624, 31771515648

Offset

0,4

Comments

All terms with n > 1 are even.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 15.

Wikipedia, Counting lattice paths

Example

. a(1) = 1:    /\  .
.
. a(3) = 2:     /\       /\
.            /\/  \     /  \/\  .
.
. a(5) = 4:
.                /\       /\         /\       /\
.             /\/  \     /  \/\   /\/  \     /  \/\
.          /\/      \ /\/      \ /      \/\ /      \/\ .

Maple

b:= proc(n, j) option remember; `if`(n=j or n=0, 1, add(
       b(n-j, i)*binomial(j-1, i-2)*i, i=1..min(j+2, n-j)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..35);

Mathematica

b[n_, j_] := b[n, j] = If[n == j || n == 0, 1, Sum[b[n - j, i]*Binomial[j - 1, i - 2]*i, {i, 1, Min[j + 2, n - j]}]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 23 2018, translated from Maple *)

Crossrefs

Column k=1 of A288318.

Cf. A000108, A281874, A287843, A287845, A287901, A287963, A287987, A289020.

Sequence in context: A098123 A066659 A343468 * A085623 A317965 A369025

Adjacent sequences: A287843 A287844 A287845 * A287847 A287848 A287849

Keyword

nonn

Author

Alois P. Heinz, Jun 01 2017

Status

approved