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A224769 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 U=(1,1), D=(1,-1) and S=(0,1).

%I A224769 #24 Dec 18 2020 12:02:06
%S A224769 1,1,2,7,33,184,1142,7629,53750,394157,2981546,23117242,182867360,
%T A224769 1470714606,11993628444,98967634147,824958769631,6937180941468,
%U A224769 58785077008641,501520244718945,4304433733010962,37142428443486254,322042675618484973,2804409601249038670
%N A224769 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 U=(1,1), D=(1,-1) and S=(0,1).
%H A224769 Alois P. Heinz, <a href="/A224769/b224769.txt">Table of n, a(n) for n = 0..1000</a>
%F A224769 a(n) ~ c * d^n / n^(3/2), where d = 3/4*(71 + 8*sqrt(2))^(1/3) + 51/(4*(71 + 8*sqrt(2))^(1/3)) + 13/4 = 9.4435356015932520820011..., c = 0.00814413508604516738631686716788556507884786... . - _Vaclav Kotesovec_, Sep 07 2014
%e A224769 a(2) = 2: UDSS, UU.
%e A224769 a(3) = 7: UDSDSSS, UDUSS, UDSSDSS, UUDSS, UDSUS, UDSSU, UUU.
%p A224769 b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1,
%p A224769       `if`(y>0, b(x, y-1)+b(x-1, y-1), 0)+b(x-1, y+1)))
%p A224769     end:
%p A224769 a:= n-> b(n, n):
%p A224769 seq(a(n), n=0..30);
%t A224769 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]]];
%t A224769 a[n_] := b[n, n];
%t A224769 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A224769 Cf. A198324 (paths to (n,0)), A225042 (with additional H-steps), A286425.
%K A224769 nonn
%O A224769 0,3
%A A224769 _Alois P. Heinz_, Apr 17 2013