A287843 Number of Dyck paths of semilength n such that each level with peaks has exactly two peaks.

1, 0, 1, 1, 2, 5, 15, 27, 76, 196, 548, 1388, 3621, 9894, 27553, 75346, 205634, 563729, 1565409, 4370226, 12191929, 33980329, 94874987, 265668404, 745652478, 2095025688, 5889310438, 16565399257, 46633521554, 131388795335, 370434641340, 1044917168292

Offset: 0

Author

Alois P. Heinz, Jun 01 2017

Keywords

• nonn

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A000108, A281874, A287845, A287846, A287963.

Column k=2 of A288108.

Programs

Maple

b:= proc(n, j) option remember; `if`(n=j or n=0, 1,
      add(b(n-j, i)*(binomial(j-1, i-1) +i*(i-1)/2*
      binomial(j-1, i-3)), i=1..min(j+3, 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]*(Binomial[j - 1, i-1] + i*(i-1)/2*Binomial[j-1, i-3]), {i, 1, Min[j + 3, n - j]}]];
a[n_] := b[n, 2];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 25 2018, translated from Maple *)