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A333105 Number of 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 A333105 #23 Oct 24 2021 04:33:08
%S A333105 1,1,2,4,9,21,51,128,331,880,2402,6724,19285,56612,169908,520723,
%T A333105 1627477,5180064,16766824,55112302,183710312,620213500,2118094664,
%U A333105 7309077920,25459737905,89438446602,316606738516,1128566016617,4048230694964,14604517154115
%N A333105 Number of 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)}.
%H A333105 Alois P. Heinz, <a href="/A333105/b333105.txt">Table of n, a(n) for n = 0..1673</a>
%H A333105 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%H A333105 Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>
%F A333105 a(n) >= A001006(n) with equality only for n <= 6.
%F A333105 a(n) ~ c * 4^n / n^(3/2), where c = 0.0019335749177095597674777855613451543338378695415042866523284... - _Vaclav Kotesovec_, Oct 24 2021
%p A333105 b:= proc(x, y) option remember; `if`(x=0, 1, add(
%p A333105       b(x-1, y+j), j=-min(1, y)..min(max(1, y), x-y-1)))
%p A333105     end:
%p A333105 a:= n-> b(n, 0):
%p A333105 seq(a(n), n=0..29);
%t A333105 b[x_, y_] := b[x, y] = If[x == 0, 1, Sum[b[x - 1, y + j],
%t A333105      {j, -Min[1, y], Min[Max[1, y], x - y - 1]}]];
%t A333105 a[n_] := b[n, 0];
%t A333105 a /@ Range[0, 29] (* _Jean-François Alcover_, Mar 30 2021, after _Alois P. Heinz_ *)
%Y A333105 Cf. A001006, A230556, A333069, A333106, A333107, A333608.
%K A333105 nonn
%O A333105 0,3
%A A333105 _Alois P. Heinz_, Mar 07 2020