A276903 Number of positive walks with n steps {-2,-1,0,1,2} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

0, 1, 2, 7, 25, 96, 382, 1567, 6575, 28096, 121847, 534953, 2373032, 10619922, 47890013, 217395690, 992640367, 4555957948, 21007405327, 97266928685, 452046424465, 2108022305795, 9860773604035, 46256877824220, 217555982625385, 1025667805621986, 4846240583558277

Offset: 0

Author

Michael Wallner, Sep 21 2016

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1437
• 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, A276901, A276902, 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, h}];
(* walks represents the number of positive walks with n steps {-h, -h+1, ... , h} that end at altitude k *)
A276903[n_] := (Do[walks[m, k, 2], {m, n}, {k, 2 m}]; walks[n, 2, 2]) (* Davin Park, Oct 10 2016 *)