A277922 Number of positive walks with n steps {-4,-3,-2,-1,1,2,3,4} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

0, 1, 3, 13, 71, 405, 2501, 15923, 104825, 704818, 4827957, 33549389, 235990887, 1676907903, 12019875907, 86804930199, 630999932585, 4613307289260, 33900874009698, 250257489686870, 1854982039556397, 13800559463237465, 103017222722691145, 771348369563479705

Offset: 0

Author

David Nguyen, Nov 04 2016

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Crossrefs

Cf. A276852, A276901, A276902, A276903, A276904.

Programs

Maple

b:= proc(n, y) option remember; `if`(n=0, `if`(y=1, 1, 0), add
     ((h-> `if`(h<1, 0, b(n-1, h)))(y+d), d=[$-4..-1, $1..4]))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Nov 12 2016

Mathematica

b[n_, y_] := b[n, y] = If[n == 0, If[y == 1, 1, 0], Sum[Function[h, If[h < 1, 0, b[n - 1, h]]][y + d], {d, Join[Range[-4, -1], Range[4]]}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 03 2017, after Alois P. Heinz *)