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A333680 Total number of nodes summed over all 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 A333680 #12 Apr 26 2021 05:22:37
%S A333680 1,2,3,8,20,48,112,272,666,1690,4367,11436,30147,80248,215550,583456,
%T A333680 1588956,4351806,11979481,33127440,91982688,256354098,716879847,
%U A333680 2010919560,5656813275,15954441334,45106324389,127809023944,362897750254,1032389760540,2942278599032
%N A333680 Total number of nodes summed over all 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.
%H A333680 Alois P. Heinz, <a href="/A333680/b333680.txt">Table of n, a(n) for n = 0..300</a>
%H A333680 Alois P. Heinz, <a href="/A333647/a333647.gif">Animation of A333647(9) = 169 paths with a(9) = 1690 nodes</a>
%H A333680 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A333680 a(n) = (n+1) * A333647(n).
%p A333680 b:= proc(x, y, t) option remember; `if`(x=0, 1, add(
%p A333680       b(x-1, y+j, j), j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
%p A333680     end:
%p A333680 a:= n-> (n+1)*b(n, 0$2):
%p A333680 seq(a(n), n=0..36);
%t A333680 b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Sum[
%t A333680      b[x-1, y+j, j], {j, Max[t-1, -y], Min[x(x-1)/2-y, t+1]}]];
%t A333680 a[n_] := (n+1) b[n, 0, 0];
%t A333680 a /@ Range[0, 36] (* _Jean-François Alcover_, Apr 26 2021, after _Alois P. Heinz_ *)
%Y A333680 Cf. A333647, A333678, A333679.
%K A333680 nonn
%O A333680 0,2
%A A333680 _Alois P. Heinz_, Apr 01 2020