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A225041 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), H=(1,0) and S=(0,1).

%I A225041 #18 May 14 2017 09:52:23
%S A225041 1,1,3,9,35,145,659,3137,15619,80177,422595,2273633,12447667,69138193,
%T A225041 388784259,2209440945,12671782579,73260414481,426545078627,
%U A225041 2499059841249,14723542302627,87181150961361,518554078448339,3097007445391441,18565515801339827
%N A225041 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), H=(1,0) and S=(0,1).
%H A225041 Alois P. Heinz, <a href="/A225041/b225041.txt">Table of n, a(n) for n = 0..1000</a>
%F A225041 a(n) ~ c * (3+2*sqrt(3))^n / n^(3/2), where c = 0.05641378816540215191327201376... . - _Vaclav Kotesovec_, Sep 07 2014
%e A225041 a(0) = 1: the empty path.
%e A225041 a(1) = 1: H.
%e A225041 a(2) = 3: HH, UD, HSD.
%e A225041 a(3) = 9: HHH, UDH, HSDH, UHD, HSHD, HUD, HHSD, UDSD, HSDSD.
%p A225041 b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1,
%p A225041        b(x-1, y)+`if`(y>0, b(x-1, y-1)+b(x, y-1), 0)+b(x-1, y+1)))
%p A225041     end:
%p A225041 a:= n-> b(n, 0):
%p A225041 seq(a(n), n=0..25);
%t A225041 b[x_, y_] := b[x, y] = If[y>x, 0, If[x==0, 1, b[x-1, y]+If[y>0, b[x-1, y-1] + b[x, y-1], 0] + b[x-1, y+1]]];
%t A225041 a[n_] := b[n, 0];
%t A225041 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 29 2017, translated from Maple *)
%Y A225041 Cf. A001006 (without S-steps), A114296 (without U-steps), A198324 (without H-steps), A225042 (paths to (n,n)), A286760.
%K A225041 nonn
%O A225041 0,3
%A A225041 _Alois P. Heinz_, Apr 25 2013