A333071 Total area under all lattice paths from (0,0) to (n,0) that do not go below the x-axis, and at (x,y) only allow steps (1,v) with v in {-1,0,1,...,y+1}.

0, 0, 1, 4, 16, 63, 239, 895, 3343, 12503, 46905, 176620, 667664, 2533699, 9650737, 36887383, 141448958, 544022417, 2098082719, 8111788699, 31434420426, 122068414186, 474932563378, 1851059631879, 7226108097869, 28250493771358, 110594307388370, 433488248791630

Offset: 0

Author

Alois P. Heinz, Mar 06 2020

Keywords

• nonn

Links

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

Crossrefs

Cf. A333069, A333070, A333107, A333504.

Programs

Maple

b:= proc(x, y) option remember; `if`(x=0, [1, 0],
      add(`if`(x+j>y, (p-> p+[0, p[1]*(y-j/2)])(
        b(x-1, y-j)), 0), j=-1-y..min(1, y)))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);

Mathematica

b[x_, y_] := b[x, y] = If[x == 0, {1, 0},
     Sum[If[x + j > y, With[{p = b[x - 1, y - j]}, p +
     {0, p[[1]] (y - j/2)}], 0], {j, -1 - y, Min[1, y]}]];
a[n_] := b[n, 0][[2]];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 05 2021, after Alois P. Heinz *)