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

0,6

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

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 *)

Crossrefs

Column k=2 of A288318.

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

Sequence in context: A271854 A077086 A010618 * A070297 A284896 A299032

Adjacent sequences: A287842 A287843 A287844 * A287846 A287847 A287848

Keyword

nonn

Author

Alois P. Heinz, Jun 01 2017

Status

approved