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A333069 Number of 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 A333069 #20 Dec 19 2020 02:28:05
%S A333069 1,1,2,4,9,22,57,155,439,1287,3886,12035,38100,122943,403410,1343321,
%T A333069 4531710,15465414,53325680,185575269,651191826,2302247822,8194892393,
%U A333069 29350405663,105713021575,382717065800,1392121894189,5085836001166,18654616951435,68678029247822
%N A333069 Number of 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 A333069 Alois P. Heinz, <a href="/A333069/b333069.txt">Table of n, a(n) for n = 0..1000</a>
%H A333069 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%H A333069 Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>
%F A333069 a(n) = A196161(n) for n = 1..8.
%F A333069 a(n) ~ c * 4^n / n^(3/2), where c = 0.0131789402414023971902275212293294628834887666310830183578424168829... - _Vaclav Kotesovec_, Mar 25 2020
%p A333069 b:= proc(x, y) option remember; `if`(x=0, 1, add(
%p A333069      `if`(x+j>y, b(x-1, y-j), 0), j=-1-y..min(1, y)))
%p A333069     end:
%p A333069 a:= n-> b(n, 0):
%p A333069 seq(a(n), n=0..33);
%t A333069 b[x_, y_] := b[x, y] = If[x == 0, 1, Sum[If[x + j > y, b[x - 1, y - j], 0], {j, -1 - y, Min[1, y]}]];
%t A333069 a[n_] := b[n, 0];
%t A333069 a /@ Range[0, 33] (* _Jean-François Alcover_, Dec 19 2020, after _Alois P. Heinz_ *)
%Y A333069 Cf. A196161, A333070, A333071, A333105, A333504.
%K A333069 nonn
%O A333069 0,3
%A A333069 _Alois P. Heinz_, Mar 06 2020