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A286427 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) and S=(0,1).

%I A286427 #16 Sep 11 2021 13:48:53
%S A286427 1,0,3,5,24,78,325,1272,5373,22572,97762,425716,1882062,8375064,
%T A286427 37601643,169773435,771096972,3518065767,16119884297,74125225732,
%U A286427 341970676466,1582133854847,7338641255894,34117910537671,158946919835352,741884929970516,3468677541274922
%N A286427 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) and S=(0,1).
%H A286427 Alois P. Heinz, <a href="/A286427/b286427.txt">Table of n, a(n) for n = 0..1000</a>
%F A286427 a(n) ~ c * 2^n * (1 + sqrt(2))^n / sqrt(n), where c = 0.0205249406642810706349830917137505197586781430984... - _Vaclav Kotesovec_, Sep 11 2021
%p A286427 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1$2],
%p A286427       (p-> p+[0, p[1]])(b(x, y-1)+b(x-1, y-1)+b(x-1, y+1))))
%p A286427     end:
%p A286427 a:= n-> b(n, 0)[2]:
%p A286427 seq(a(n), n=0..30);
%Y A286427 Cf. A198324.
%K A286427 nonn
%O A286427 0,3
%A A286427 _Alois P. Heinz_, May 14 2017