%I #17 Dec 22 2020 03:54:50

%S 1,2,6,16,45,131,393,1218,3887,12736,42707,146113,508610,1796848,

%T 6428953,23253209,84893617,312435085,1157899672,4317354453,

%U 16183476500,60947573729,230481995102,874810511970,3331322503398,12723257204883,48722782351656,187028551724723

%N Total number of nodes summed over all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.

%H Alois P. Heinz, <a href="/A337318/b337318.txt">Table of n, a(n) for n = 0..1666</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%F a(n) ~ c * 4^n / sqrt(n), where c = 0.028711801689489498782112731663771630297082311282971968906589032765122715... - _Vaclav Kotesovec_, Oct 24 2020

%p b:= proc(x, y) option remember; `if`(x=0, [1$2],

%p add(add((p-> p+[0, p[1]])(b(x-h, y-v)), h=1..

%p min(x-y+v, max(1, y-v))), v=-1..min(y, 1)))

%p end:

%p a:= n-> b(n, 0)[2]:

%p seq(a(n), n=0..30);

%t b[x_, y_] := b[x, y] = If[x == 0, {1, 1},

%t Sum[Sum[Function[p, p + {0, p[[1]]}][b[x - h, y - v]], {h, 1,

%t Min[x - y + v, Max[1, y - v]]}], {v, -1, Min[y, 1]}]];

%t a[n_] := b[n, 0][[2]];

%t a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 22 2020, after _Alois P. Heinz_ *)

%Y Cf. A337067, A337863.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Oct 12 2020