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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.

%I A333647 #21 Apr 26 2021 05:23:03
%S A333647 1,1,1,2,4,8,16,34,74,169,397,953,2319,5732,14370,36466,93468,241767,
%T A333647 630499,1656372,4380128,11652459,31168689,83788315,226272531,
%U A333647 613632359,1670604607,4564607998,12513715526,34412992018,94912212872,262484672621,727770127583
%N 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.
%C A333647 The maximal height in all paths of length n is floor(ceil(n/2)^2/4) = A008642(n-3) for n>2.
%H A333647 Alois P. Heinz, <a href="/A333647/b333647.txt">Table of n, a(n) for n = 0..300</a>
%H A333647 Alois P. Heinz, <a href="/A333647/a333647.gif">Animation of a(9) = 169 paths</a>
%H A333647 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A333647 b:= proc(x, y, t) option remember; `if`(x=0, 1, add(
%p A333647       b(x-1, y+j, j), j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
%p A333647     end:
%p A333647 a:= n-> b(n, 0$2):
%p A333647 seq(a(n), n=0..40);
%t A333647 b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Sum[
%t A333647      b[x-1, y+j, j], {j, Max[t-1, -y], Min[x(x-1)/2-y, t+1]}]];
%t A333647 a[n_] := b[n, 0, 0];
%t A333647 a /@ Range[0, 40] (* _Jean-François Alcover_, Apr 26 2021, after _Alois P. Heinz_ *)
%Y A333647 Cf. A008642, A225338, A333678, A333679, A333680, A333682.
%K A333647 nonn
%O A333647 0,4
%A A333647 _Alois P. Heinz_, Mar 31 2020