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A333608 Sum of the heights of all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (1,v) with v in {-1,0,...,max(y,1)}.

%I A333608 #16 May 12 2020 08:41:06
%S A333608 0,0,1,3,9,25,70,200,584,1742,5304,16471,52120,167885,549856,1828897,
%T A333608 6170108,21087458,72923515,254880303,899454849,3201729220,11486266036,
%U A333608 41497996004,150879471934,551723923040,2027990653855,7489507917594,27777837416779,103427750936183
%N A333608 Sum of the heights of all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (1,v) with v in {-1,0,...,max(y,1)}.
%C A333608 The maximal height in all paths of length n is A103354(n-1).
%H A333608 Alois P. Heinz, <a href="/A333608/b333608.txt">Table of n, a(n) for n = 0..500</a>
%H A333608 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%H A333608 Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>
%p A333608 b:= proc(x, y, h) option remember;
%p A333608      `if`(x=0, h, add(b(x-1, y+j, max(y, h)),
%p A333608       j=-min(1, y)..min(max(1, y), x-y-1)))
%p A333608     end:
%p A333608 a:= n-> b(n, 0$2):
%p A333608 seq(a(n), n=0..29);
%t A333608 b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[b[x - 1, y + j, Max[y, h]], {j, -Min[1, y], Min[Max[1, y], x - y - 1]}]];
%t A333608 a[n_] := b[n, 0, 0];
%t A333608 a /@ Range[0, 29] (* _Jean-François Alcover_, May 12 2020, after Maple *)
%Y A333608 Cf. A103354, A333105, A333106, A333107, A333498, A333504.
%K A333608 nonn
%O A333608 0,4
%A A333608 _Alois P. Heinz_, Mar 28 2020