A337863 Twice the total area of all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.

0, 0, 2, 8, 32, 123, 459, 1715, 6432, 24256, 91993, 350817, 1344559, 5176106, 20003683, 77567894, 301667089, 1176204195, 4596267116, 17995972170, 70581437130, 277244854010, 1090484851145, 4294324521033, 16929104620229, 66801975748076, 263827090563129

Offset: 0

Author

Alois P. Heinz, Oct 12 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1663

Crossrefs

Cf. A337067, A337318.

Programs

Maple

b:= proc(x, y) option remember; `if`(x=0, [1, 0],
      add(add((p-> p+[0, p[1]*h*(2*y-v)])(b(x-h, y-v)),
      h=1..min(x-y+v, max(1, y-v))), v=-1..min(y, 1)))
    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[Sum[Function[p, p + {0, p[[1]]*h*(2*y - v)}][b[x - h, y - v]],
     {h, 1, Min[x - y + v, Max[1, y - v]]}], {v, -1, Min[y, 1]}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 24 2022, after Alois P. Heinz *)