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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.
6
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
0,4
COMMENTS
The maximal height in all paths of length n is floor(ceil(n/2)^2/4) = A008642(n-3) for n>2.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Mar 31 2020
STATUS
approved