A287963 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has one or two peaks.

1, 1, 1, 2, 5, 10, 28, 71, 194, 532, 1495, 4256, 12176, 35251, 102664, 300260, 881909, 2599948, 7688164, 22788527, 67676144, 201308938, 599676445, 1788564038, 5339905904, 15956230705, 47713265536, 142763240666, 427390085963, 1280058256294, 3835332884686

Offset: 0

Author

Alois P. Heinz, Jun 03 2017

Keywords

• nonn

Examples

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

Links

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

Crossrefs

Cf. A000108, A281874, A287843, A287845, A287846.

Programs

Maple

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

Mathematica

b[n_, j_] := b[n, j] = If[n == j, 1, Sum[b[n - j, i]*i*(Binomial[j - 1, i - 2] + (i - 1)/2*Binomial[j - 1, i - 3]), {i, 2, Min[j + 3, n - j]}]];
a[n_] := If[n == 0, 1, b[n, 1] + b[n, 2]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 29 2018, from Maple *)