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A198324 Number of 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 A198324 #30 May 14 2017 09:07:19
%S A198324 1,0,1,1,4,10,35,116,427,1584,6146,24216,97754,400080,1662645,6986127,
%T A198324 29669872,127101015,548839687,2386211664,10439207266,45920497075,
%U A198324 203004397362,901459381683,4019351034816,17987665701788,80773320086286,363842478143834
%N A198324 Number of 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 A198324 Alois P. Heinz, <a href="/A198324/b198324.txt">Table of n, a(n) for n = 0..1000</a>
%F A198324 a(n) ~ c * (2*(1+sqrt(2)))^n / n^(3/2), where c = 0.01202323187423280845930143205554758... . - _Vaclav Kotesovec_, Sep 07 2014
%e A198324 a(4) = 4: UDSDSD, UDUD, UDSSDD, UUDD.
%e A198324 a(5) = 10: UDSDSDSD, UDUDSD, UDSSDDSD, UUDDSD, UDSDUD, UDSDSSDD, UDUSDD, UDSSDSDD, UUDSDD, UDSUDD.
%p A198324 b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1,
%p A198324       `if`(y>0, b(x, y-1)+b(x-1, y-1), 0)+b(x-1, y+1)))
%p A198324     end:
%p A198324 a:= n-> b(n, 0):
%p A198324 seq(a(n), n=0..30);
%t A198324 b[x_, y_] := b[x, y] = If[y>x, 0, If[x == 0, 1, If[y>0, b[x, y-1] + b[x-1, y-1], 0] + b[x-1, y+1]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 05 2015, after _Alois P. Heinz_ *)
%Y A198324 Cf. A000108 (without S-steps), A224769 (paths to (n,n)), A225041 (with additional H-steps), A286427.
%K A198324 nonn
%O A198324 0,5
%A A198324 _Alois P. Heinz_, Apr 18 2013