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A286760 Total number of nodes summed over all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps U=(1,1), D=(1,-1), H=(1,0) and S=(0,1).

%I A286760 #10 Jun 28 2022 03:27:47
%S A286760 1,2,10,42,214,1098,5978,33190,189078,1093490,6414714,38027030,
%T A286760 227489950,1370980490,8314674202,50696630838,310541818382,
%U A286760 1909850054666,11786947172234,72969941803662,452976340653030,2818815920369754,17579546535174946,109850944544149134
%N A286760 Total number of nodes summed over all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps U=(1,1), D=(1,-1), H=(1,0) and S=(0,1).
%H A286760 Alois P. Heinz, <a href="/A286760/b286760.txt">Table of n, a(n) for n = 0..1000</a>
%F A286760 a(n) ~ c * (3 + 2*sqrt(3))^n / sqrt(n), where c = 0.0889843039487036085233000284915570190371055498671732340656... - _Vaclav Kotesovec_, Sep 11 2021
%p A286760 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1$2],
%p A286760       (p-> p+[0, p[1]])(b(x-1, y)+b(x, y-1)+b(x-1, y-1)+b(x-1, y+1))))
%p A286760     end:
%p A286760 a:= n-> b(n, 0)[2]:
%p A286760 seq(a(n), n=0..30);
%t A286760 b[x_, y_] := b[x, y] = If[y<0 || y>x, 0, If[x == 0, {1, 1}, Function[
%t A286760    p, p+{0, p[[1]]}][b[x-1, y] + b[x, y-1] + b[x-1, y-1] + b[x-1, y+1]]]];
%t A286760 a[n_] := b[n, 0][[2]];
%t A286760 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 28 2022, after _Alois P. Heinz_ *)
%Y A286760 Cf. A176394, A225041.
%K A286760 nonn
%O A286760 0,2
%A A286760 _Alois P. Heinz_, May 14 2017