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A337318 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}.

%I A337318 #17 Dec 22 2020 03:54:50
%S A337318 1,2,6,16,45,131,393,1218,3887,12736,42707,146113,508610,1796848,
%T A337318 6428953,23253209,84893617,312435085,1157899672,4317354453,
%U A337318 16183476500,60947573729,230481995102,874810511970,3331322503398,12723257204883,48722782351656,187028551724723
%N A337318 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 A337318 Alois P. Heinz, <a href="/A337318/b337318.txt">Table of n, a(n) for n = 0..1666</a>
%H A337318 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A337318 a(n) ~ c * 4^n / sqrt(n), where c = 0.028711801689489498782112731663771630297082311282971968906589032765122715... - _Vaclav Kotesovec_, Oct 24 2020
%p A337318 b:= proc(x, y) option remember; `if`(x=0, [1$2],
%p A337318       add(add((p-> p+[0, p[1]])(b(x-h, y-v)), h=1..
%p A337318       min(x-y+v, max(1, y-v))), v=-1..min(y, 1)))
%p A337318     end:
%p A337318 a:= n-> b(n, 0)[2]:
%p A337318 seq(a(n), n=0..30);
%t A337318 b[x_, y_] := b[x, y] = If[x == 0, {1, 1},
%t A337318     Sum[Sum[Function[p, p + {0, p[[1]]}][b[x - h, y - v]], {h, 1,
%t A337318     Min[x - y + v, Max[1, y - v]]}], {v, -1, Min[y, 1]}]];
%t A337318 a[n_] := b[n, 0][[2]];
%t A337318 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 22 2020, after _Alois P. Heinz_ *)
%Y A337318 Cf. A337067, A337863.
%K A337318 nonn
%O A337318 0,2
%A A337318 _Alois P. Heinz_, Oct 12 2020