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A286425 Total number of nodes summed over all lattice paths from (0,0) to (n,n) 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 A286425 #21 Sep 11 2021 13:48:14
%S A286425 1,2,8,44,285,2028,15338,120960,983108,8172094,69116592,592590616,
%T A286425 5136777504,44928712804,395907022448,3510622573064,31296093794827,
%U A286425 280275392413204,2520017580255461,22736733105613548,205767848345966976,1867240544055742660
%N A286425 Total number of nodes summed over all lattice paths from (0,0) to (n,n) 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 A286425 Alois P. Heinz, <a href="/A286425/b286425.txt">Table of n, a(n) for n = 0..1000</a>
%F A286425 a(n) ~ c * d^n / sqrt(n), where d = (3*(71 + 8*sqrt(2))^(1/3))/4 + 51/(4*(71 + 8*sqrt(2))^(1/3)) + 13/4 = 9.443535601593252082001105527294087383986236797... and c = 0.0201623254316291127574085659620180015446126055020315052104102916... - _Vaclav Kotesovec_, Sep 11 2021
%p A286425 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1$2],
%p A286425       (p-> p+[0, p[1]])(b(x, y-1)+b(x-1, y-1)+b(x-1, y+1))))
%p A286425     end:
%p A286425 a:= n-> b(n$2)[2]:
%p A286425 seq(a(n), n=0..30);
%Y A286425 Cf. A224769.
%K A286425 nonn
%O A286425 0,2
%A A286425 _Alois P. Heinz_, May 14 2017