A333498 Sum of the heights of all Motzkin paths of length n.

0, 0, 1, 3, 9, 25, 70, 196, 552, 1560, 4423, 12573, 35826, 102310, 292786, 839554, 2411945, 6941593, 20011328, 57779038, 167069317, 483739961, 1402413161, 4070537585, 11827842021, 34403798725, 100167396088, 291903951462, 851380987390, 2485175809878

Offset

0,4

Links

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

Wikipedia, Motzkin number

Formula

a(n) = Sum_{k=1..floor(n/2)} k * A097862(n,k).

Maple

b:= proc(x, y, h) option remember; `if`(x=0, h, 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);

Mathematica

b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[b[x - 1, y + j, Max[h, y]], {j, -Min[1, y], Min[1, x - y - 1]}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 35] (* Jean-François Alcover, May 10 2020, after Maple *)

Crossrefs

Cf. A001006, A057585, A097862, A136439, A333504, A333608.

Sequence in context: A007046 A211286 A211285 * A097861 A211284 A211283

Adjacent sequences: A333495 A333496 A333497 * A333499 A333500 A333501

Keyword

nonn

Author

Alois P. Heinz, Mar 24 2020

Status

approved