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

0,5

Links

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

Wikipedia, Counting lattice paths

Example

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

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 *)

Crossrefs

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

Column k=2 of A288108.

Sequence in context: A290527 A146117 A290550 * A290828 A290836 A191313

Adjacent sequences: A287840 A287841 A287842 * A287844 A287845 A287846

Keyword

nonn

Author

Alois P. Heinz, Jun 01 2017

Status

approved