A276901 Number of positive walks with n steps {-3,-2,-1,1,2,3} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

0, 1, 2, 9, 34, 159, 730, 3579, 17762, 90538, 467796, 2452727, 12997554, 69549847, 375159290, 2038068813, 11140256754, 61227097438, 338140106124, 1875581756078, 10444142352812, 58364192607047, 327203347219250, 1839778650617309, 10372512509521074

Offset: 0

Author

Michael Wallner, Sep 21 2016

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1291
• 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 preprint arXiv:1609.06473 [math.CO], 2016.

Crossrefs

Cf. A276852, A276902, A276903, A276904.

Programs

Mathematica

walks[n_, k_, h_] = 0;
walks[1, k_, h_] := Boole[0 < k <= h];
walks[n_, k_, h_] /; n >= 2 && k > 0 := walks[n, k, h] = Sum[walks[n - 1, k - x, h], {x, h}] + Sum[walks[n - 1, k + x, h], {x, h}];
(* walks represents the number of positive walks with n steps {-h, -h+1, ... -1, 1, ..., h} that end at altitude k *)
A276901[n_] := (Do[walks[m, k, 3], {m, n}, {k, 3 m}]; walks[n, 2, 3]) (* Davin Park, Oct 10 2016 *)