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A333504 Sum of the heights of all lattice paths from (0,0) to (n,0) that do not go below the x-axis, and at (x,y) only allow steps (1,v) with v in {-1,0,1,...,y+1}.

%I A333504 #13 Apr 26 2021 05:23:12
%S A333504 0,0,1,3,9,28,88,282,921,3058,10302,35159,121406,423704,1493046,
%T A333504 5307276,19014642,68609686,249149529,910000728,3341113126,12325295866,
%U A333504 45664033813,169846998495,634020229888,2374550269819,8920273989351,33604033638696,126919824985533
%N A333504 Sum of the heights of all lattice paths from (0,0) to (n,0) that do not go below the x-axis, and at (x,y) only allow steps (1,v) with v in {-1,0,1,...,y+1}.
%H A333504 Alois P. Heinz, <a href="/A333504/b333504.txt">Table of n, a(n) for n = 0..400</a>
%H A333504 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A333504 b:= proc(x, y, h) option remember; `if`(x=0, h, add((t->
%p A333504      `if`(x>t, b(x-1, t, max(h, t)), 0))(y-j), j=-1-y..min(1, y)))
%p A333504     end:
%p A333504 a:= n-> b(n, 0$2):
%p A333504 seq(a(n), n=0..33);
%t A333504 b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[With[{t = y - j},
%t A333504      If[x > t, b[x - 1, t, Max[h, t]], 0]], {j, -1 - y, Min[1, y]}]];
%t A333504 a[n_] := b[n, 0, 0];
%t A333504 a /@ Range[0, 33] (* _Jean-François Alcover_, Apr 26 2021, after _Alois P. Heinz_ *)
%Y A333504 Cf. A333069, A333070, A333071, A333498, A333608.
%K A333504 nonn
%O A333504 0,4
%A A333504 _Alois P. Heinz_, Mar 24 2020