A372033 The total number of levels visited by all Motzkin paths of length n.

1, 1, 3, 7, 18, 46, 121, 323, 875, 2395, 6611, 18371, 51337, 144145, 406420, 1150126, 3265412, 9298372, 26547710, 75978322, 217921336, 626287520, 1803176384, 5200298000, 15020569818, 43447201226, 125837214564, 364911724264, 1059404265599, 3078918594707

Offset: 0

Author

Alois P. Heinz, Apr 16 2024

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Motzkin number

Crossrefs

Row sums of A372014.

Cf. A001006, A261003, A333498.

Programs

Maple

b:= proc(x, y, h) option remember; `if`(x=0, h+1, add(
      b(x-1, y+j, max(h, y)), j=-min(1, y)..min(1, x-y-1)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..35);

Formula

a(n) = A001006(n) + A333498(n).