A333647 Number of nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.
1, 1, 1, 2, 4, 8, 16, 34, 74, 169, 397, 953, 2319, 5732, 14370, 36466, 93468, 241767, 630499, 1656372, 4380128, 11652459, 31168689, 83788315, 226272531, 613632359, 1670604607, 4564607998, 12513715526, 34412992018, 94912212872, 262484672621, 727770127583
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..300
- Alois P. Heinz, Animation of a(9) = 169 paths
- Wikipedia, Counting lattice paths
Programs
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Maple
b:= proc(x, y, t) option remember; `if`(x=0, 1, add( b(x-1, y+j, j), j=max(t-1, -y)..min(x*(x-1)/2-y, t+1))) end: a:= n-> b(n, 0$2): seq(a(n), n=0..40); -
Mathematica
b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Sum[ b[x-1, y+j, j], {j, Max[t-1, -y], Min[x(x-1)/2-y, t+1]}]]; a[n_] := b[n, 0, 0]; a /@ Range[0, 40] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)
Comments