A287845 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has exactly two peaks.

1, 0, 1, 0, 0, 3, 6, 0, 9, 54, 138, 207, 360, 1368, 4545, 11304, 25182, 61605, 173916, 498798, 1347417, 3497328, 9147060, 24630669, 67414590, 184065966, 498495303, 1345622436, 3642036804, 9900361107, 26982011250, 73570082760, 200540053395, 546660151722

Offset: 0

Author

Alois P. Heinz, Jun 01 2017

Keywords

• nonn

Examples

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

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=2 of A288318.

Cf. A000108, A281874, A287843, A287846, A287963, A287987.

Programs

Maple

b:= proc(n, j) option remember;
      `if`(n=j or n=0, 1, add(b(n-j, i)*i*(i-1)/2
       *binomial(j-1, i-3), i=3..min(j+2, n-j)))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..35);

Mathematica

b[n_, j_] := b[n, j] = If[n == j || n == 0, 1, Sum[b[n - j, i]*i*(i - 1)/2* Binomial[j - 1, i - 3], {i, 3, Min[j + 2, n - j]}]];
a[n_] := b[n, 2];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 25 2018, translated from Maple *)