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A333106 Total number of nodes summed over 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 A333106 #16 Oct 24 2021 04:41:03
%S A333106 1,2,6,16,45,126,357,1024,2979,8800,26422,80688,250705,792568,2548620,
%T A333106 8331568,27667109,93241152,318569656,1102246040,3857916552,
%U A333106 13644697000,48716177272,175417870080,636493447625,2325399611652,8548381939932,31599848465276
%N A333106 Total number of nodes summed over 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)}.
%H A333106 Alois P. Heinz, <a href="/A333106/b333106.txt">Table of n, a(n) for n = 0..1668</a>
%H A333106 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%H A333106 Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>
%F A333106 a(n) ~ c * 4^n / sqrt(n), where c = 0.0019335749177095597674777855613451543338378695415042866523284... - _Vaclav Kotesovec_, Oct 24 2021
%p A333106 b:= proc(x, y) option remember; `if`(x=0, 1, add(
%p A333106       b(x-1, y+j), j=-min(1, y)..min(max(1, y), x-y-1)))
%p A333106     end:
%p A333106 a:= n-> (n+1)*b(n, 0):
%p A333106 seq(a(n), n=0..29);
%t A333106 b[x_, y_] := b[x, y] = If[x == 0, 1,
%t A333106      Sum[b[x-1, y+j], {j, -Min[1, y], Min[Max[1, y], x-y-1]}]];
%t A333106 a[n_] := (n+1) b[n, 0];
%t A333106 a /@ Range[0, 29] (* _Jean-François Alcover_, Apr 05 2021, after _Alois P. Heinz_ *)
%Y A333106 Cf. A333070, A333105, A333107, A333608.
%K A333106 nonn
%O A333106 0,2
%A A333106 _Alois P. Heinz_, Mar 07 2020