A333107 Total area under all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (1,v) with v in {-1,0,...,max(y,1)}.

0, 0, 1, 4, 16, 56, 190, 637, 2131, 7156, 24215, 82758, 285991, 999715, 3534394, 12631420, 45601759, 166169360, 610650687, 2261234467, 8430749631, 31625520000, 119281312293, 452077280484, 1720796968459, 6575385383602, 25212139233077, 96970372087853

Offset: 0

Author

Alois P. Heinz, Mar 07 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1665
• Wikipedia, Counting lattice paths
• Wikipedia, Motzkin number

Crossrefs

Cf. A333071, A333105, A333106, A333608.

Programs

Maple

b:= proc(x, y) option remember; `if`(x=0, [1, 0], add(
      (p-> p+[0, p[1]*(y+j/2)])(b(x-1, y+j)),
       j=-min(1, y)..min(max(1, y), x-y-1)))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..29);

Mathematica

b[x_, y_] := b[x, y] = If[x == 0, {1, 0}, Sum[
     Function[p, p + {0, p[[1]]*(y + j/2)}][b[x - 1, y + j]],
     {j, -Min[1, y], Min[Max[1, y], x - y - 1]}]];
a[n_] :=  b[n, 0][[2]];
a /@ Range[0, 29] (* Jean-François Alcover, Apr 05 2021, after Alois P. Heinz *)