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

Author

Alois P. Heinz, Jun 01 2017

Keywords

• nonn

Comments

All terms with n > 1 are even.

Examples

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

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

Crossrefs

Column k=1 of A288318.

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

Programs

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 *)