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%I #18 Jul 23 2017 03:03:05
%S 0,0,0,4,28,157,820,4155,20742,102725,506504,2491230,12236520,
%T 60063399,294748884,1446436680,7099442700,34855583275,171187439920,
%U 841084246980,4134129246180,20328683526575,100003531112300,492153054177155
%N Number of UUU's in all skew Dyck paths of semilength n.
%C A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps.
%H Vincenzo Librandi, <a href="/A128721/b128721.txt">Table of n, a(n) for n = 0..300</a>
%H E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
%F a(n) = Sum_{k=0..n-2} k*A128719(n,k) (n >= 2).
%F G.f.: (2zg - g + 1 - z + z^2)/(2zg + z - 1), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
%F Recurrence: 2*(n+1)*(121*n-348)*a(n) = (1663*n^2 - 4620*n + 1392)*a(n-1) - (2476*n^2 - 11133*n + 11787)*a(n-2) + 5*(n-4)*(211*n-537)*a(n-3). - _Vaclav Kotesovec_, Nov 19 2012
%F a(n) ~ 9*5^(n-3/2)/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Nov 19 2012
%e a(3)=4 because each of the paths UUUDDD, UUUDLD, UUUDDL and UUUDLL contains one UUU, while the other six paths of semilength 3 contain no UUU's.
%p G:=(1-5*z+4*z^2-2*z^3-(1-2*z)*sqrt(1-6*z+5*z^2))/2/z/sqrt(1-6*z+5*z^2): Gser:=series(G,z=0,28): seq(coeff(Gser,z,n),n=0..25);
%t CoefficientList[Series[(2*x*(1-x-Sqrt[1-6*x+5*x^2])/(2*x)-(1-x-Sqrt[1-6*x+5*x^2])/(2*x)+1-x+x^2)/(2*x*(1-x-Sqrt[1-6*x+5*x^2])/(2*x)+x-1), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Nov 19 2012 *)
%Y Cf. A128719.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Mar 30 2007
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