A276852 Number of positive walks with n steps {-3,-2,-1,1,2,3} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

0, 1, 2, 7, 28, 121, 560, 2677, 13230, 66742, 343092, 1788681, 9439870, 50321865, 270594896, 1465941763, 7993664588, 43839212778, 241650560756, 1338084935826, 7439615051328, 41516113036777, 232452845782308, 1305500166481715, 7352433083806020, 41514430735834714

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1292

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.

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 *)
A276852[n_] := (Do[walks[m, k, 3], {m, n}, {k, 3 m}]; walks[n, 1, 3]) (* Davin Park, Oct 10 2016 *)

Crossrefs

Cf. A276901, A276902, A276903, A276904.

Sequence in context: A241371 A368970 A010683 * A352861 A293266 A150659

Adjacent sequences: A276849 A276850 A276851 * A276853 A276854 A276855

Keyword

nonn,walk

Author

Michael Wallner, Sep 21 2016

Status

approved