A277920 Number of positive walks with n steps {-4,-3,-2,-1,0,1,2,3,4} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

0, 1, 4, 20, 120, 780, 5382, 38638, 285762, 2162033, 16655167, 130193037, 1030117023, 8234025705, 66391916397, 539360587341, 4410492096741, 36274113675369, 299864297741292, 2490192142719336, 20764402240048267, 173784940354460219, 1459360304511145146

Offset: 0

Author

David Nguyen, Nov 04 2016

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1054
• 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. A276902, A276852, A276901, 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..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, -4, 4}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 03 2017, after Alois P. Heinz *)