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A128746 Height of the last peak summed over all skew Dyck paths of semilength n.

%I A128746 #17 Mar 22 2017 03:36:28
%S A128746 1,5,22,94,401,1723,7475,32749,144803,645627,2900256,13115820,
%T A128746 59669295,272918415,1254314310,5789850730,26831078075,124785337255,
%U A128746 582247766810,2724905891890,12787603121195,60162698218325,283715348775727
%N A128746 Height of the last peak summed over all skew Dyck paths of semilength n.
%C A128746 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 the path is defined to be the number of its steps.
%H A128746 G. C. Greubel, <a href="/A128746/b128746.txt">Table of n, a(n) for n = 1..1000</a>
%H A128746 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 A128746 a(n) = Sum_{k=1,..,n} A128745(n,k).
%F A128746 G.f.: 2*z*(1+z+sqrt(1-6*z+5*z^2))/(1-3*z+sqrt(1-6*z+5*z^2))^2.
%F A128746 a(n) ~ 5^(n+3/2)/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F A128746 Conjecture: -(n+2)*(n-1)*a(n) +(6*n^2-3*n+2)*a(n-1) -5*n*(n-2)*a(n-2)=0. - _R. J. Mathar_, Aug 08 2015
%e A128746 a(2)=5 because the skew Dyck paths of semilength 2 are UD(UD), U(UD)D and U(UD)L and their last peaks (shown between parentheses) have heights 1, 2 and 2, respectively.
%p A128746 G:=2*z*(1+z+sqrt(1-6*z+5*z^2))/(1-3*z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=1..27);
%t A128746 Rest[CoefficientList[Series[2*x*(1+x+Sqrt[1-6*x+5*x^2])/(1-3*x+Sqrt[1-6*x+5*x^2])^2, {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A128746 (PARI) z='z+O('z^50); Vec(2*z*(1+z+sqrt(1-6*z+5*z^2))/(1-3*z + sqrt(1-6*z+5*z^2))^2) \\ _G. C. Greubel_, Mar 20 2017
%Y A128746 Cf. A128745.
%K A128746 nonn
%O A128746 1,2
%A A128746 _Emeric Deutsch_, Mar 31 2007