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