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A128743 Number of UU's (i.e., doublerises) in all skew Dyck paths of semilength n.

%I A128743 #22 Dec 26 2017 03:22:47
%S A128743 0,0,2,13,69,346,1700,8286,40264,195488,949302,4613025,22436997,
%T A128743 109240038,532410060,2597468685,12684628125,62002335160,303332650190,
%U A128743 1485213237135,7277719953415,35687662907750,175120787451540
%N A128743 Number of UU's (i.e., doublerises) in all skew Dyck paths of semilength n.
%C A128743 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 A128743 G. C. Greubel, <a href="/A128743/b128743.txt">Table of n, a(n) for n = 0..1000</a>
%H A128743 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 A128743 a(n) = Sum_{k=0..n-1} k*A128718(n,k).
%F A128743 G.f.: (1-4*z+z^2+(z-1)*sqrt(1-6*z+5*z^2))/(2*z*sqrt(1-6*z+5*z^2)).
%F A128743 a(n) ~ 3*5^(n-1/2)/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014
%F A128743 Conjecture: (n+1)*(n-2)^2*a(n) -(n-1)*(6*n^2-15*n+4)*a(n-1) +5*(n-2)*(n-1)^2*a(n-2)=0. - _R. J. Mathar_, Jun 17 2016
%F A128743 Conjecture verified using the differential equation 4*g(z)+(20*z^3+2*z^2-2*z)*g'(z)+(25*z^4-15*z^3)*g''(z)+(5*z^5-6*z^4+z^3)*g'''(z)=0 satisfied by the G.f. - _Robert Israel_, Dec 25 2017
%e A128743 a(2)=2 because the paths of semilength 2 are UDUD, UUDD and UUDL, having altogether 2 UU's.
%p A128743 G:=(1-4*z+z^2+(z-1)*sqrt(1-6*z+5*z^2))/2/z/sqrt(1-6*z+5*z^2): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..25);
%t A128743 CoefficientList[Series[(1-4*x+x^2+(x-1)*Sqrt[1-6*x+5*x^2])/2/x/Sqrt[1-6*x+5*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A128743 (PARI) z='z+O('z^50); concat([0,0], Vec((1-4*z+z^2+(z-1)*sqrt(1-6*z+5*z^2))/(2*z*sqrt(1-6*z+5*z^2)))) \\ _G. C. Greubel_, Mar 20 2017
%Y A128743 Cf. A128718.
%K A128743 nonn
%O A128743 0,3
%A A128743 _Emeric Deutsch_, Mar 30 2007