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A333107 Total area under 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 A333107 #15 Apr 05 2021 09:18:41
%S A333107 0,0,1,4,16,56,190,637,2131,7156,24215,82758,285991,999715,3534394,
%T A333107 12631420,45601759,166169360,610650687,2261234467,8430749631,
%U A333107 31625520000,119281312293,452077280484,1720796968459,6575385383602,25212139233077,96970372087853
%N A333107 Total area under 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 A333107 Alois P. Heinz, <a href="/A333107/b333107.txt">Table of n, a(n) for n = 0..1665</a>
%H A333107 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%H A333107 Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>
%p A333107 b:= proc(x, y) option remember; `if`(x=0, [1, 0], add(
%p A333107       (p-> p+[0, p[1]*(y+j/2)])(b(x-1, y+j)),
%p A333107        j=-min(1, y)..min(max(1, y), x-y-1)))
%p A333107     end:
%p A333107 a:= n-> b(n, 0)[2]:
%p A333107 seq(a(n), n=0..29);
%t A333107 b[x_, y_] := b[x, y] = If[x == 0, {1, 0}, Sum[
%t A333107      Function[p, p + {0, p[[1]]*(y + j/2)}][b[x - 1, y + j]],
%t A333107      {j, -Min[1, y], Min[Max[1, y], x - y - 1]}]];
%t A333107 a[n_] :=  b[n, 0][[2]];
%t A333107 a /@ Range[0, 29] (* _Jean-François Alcover_, Apr 05 2021, after _Alois P. Heinz_ *)
%Y A333107 Cf. A333071, A333105, A333106, A333608.
%K A333107 nonn
%O A333107 0,4
%A A333107 _Alois P. Heinz_, Mar 07 2020