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A224776 Number of 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 D=(1,-1), H=(1,0) and S=(0,1).

%I A224776 #26 Jun 28 2022 03:26:31
%S A224776 1,1,3,14,83,568,4271,34296,288946,2524676,22695611,208713400,
%T A224776 1955285936,18601484936,179267898087,1746795785272,17183086302528,
%U A224776 170427862676296,1702621483524154,17118538010217472,173092651634957516,1759113081143064184,17959329720442879275
%N A224776 Number of 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 D=(1,-1), H=(1,0) and S=(0,1).
%H A224776 Alois P. Heinz, <a href="/A224776/b224776.txt">Table of n, a(n) for n = 0..960</a>
%F A224776 a(n) ~ c * ((11+5*sqrt(5))/2)^n / n^(3/2), where c = 0.01403940208697420741365874329992235342402687... . - _Vaclav Kotesovec_, Sep 07 2014
%e A224776 a(0) = 1: the empty path.
%e A224776 a(1) = 1: HS.
%e A224776 a(2) = 3: HSHS, HHSS, HSDSS.
%e A224776 a(3) = 14: HSHSHS, HHSSHS, HSDSSHS, HSHHSS, HHSHSS, HSDSHSS, HHHSSS, HSDHSSS, HSHDSSS, HHSDSSS, HSDSDSSS, HSHSDSS, HHSSDSS, HSDSSDSS.
%p A224776 b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1,
%p A224776        b(x-1, y)+`if`(y>0, b(x, y-1), 0)+b(x-1, y+1)))
%p A224776     end:
%p A224776 a:= n-> b(n, n):
%p A224776 seq(a(n), n=0..25);
%t A224776 b[x_, y_] := b[x, y] = If[y > x, 0, If[x == 0, 1, b[x - 1, y] + If[y > 0, b[x, y - 1], 0] + b[x - 1, y + 1]]];
%t A224776 a[n_] := b[n, n];
%t A224776 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A224776 Cf. A000108 (without D-steps), A114296 (paths to (n,0)), A225042 (with additional U-steps), A244593, A286764.
%K A224776 nonn
%O A224776 0,3
%A A224776 _Alois P. Heinz_, Apr 25 2013