A348202 Number of nonnegative lattice paths from (0,0) to (n,0) using steps in {(1,-4), (1,-1), (1,0), (1,1)}.

1, 1, 2, 4, 9, 22, 57, 155, 435, 1249, 3645, 10770, 32143, 96747, 293359, 895373, 2748803, 8483035, 26302248, 81896176, 255967640, 802790415, 2525691721, 7968972542, 25209580699, 79942927651, 254077293876, 809192984902, 2582113984084, 8254273128869

Offset: 0

Author

Alois P. Heinz, Oct 06 2021

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1907
• Alois P. Heinz, Animation of a(7) = 155 paths
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A000108, A001006, A025235, A036765, A333069, A333105, A337067.

Programs

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y-j), j=[-4, -1, 0, 1])))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..31);

Mathematica

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y - j], {j, {-4, -1, 0, 1}}]]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 3.3640233336410979391691803264403704977... is the root of the equation 256*d^5 - 1280*d^4 + 960*d^3 + 2267*d^2 - 1324*d - 4112 = 0 and c = 0.710307351107763693658610320440791667652705027171696102847138... - Vaclav Kotesovec, Oct 24 2021