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%I #16 Mar 20 2017 03:41:46
%S 0,0,0,1,7,38,192,946,4616,22440,108964,529133,2571079,12504038,
%T 60872038,296641049,1447054867,7065841496,34534088328,168933369259,
%U 827073303197,4052396628306,19870029768028,97495408609784
%N Number of LL'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 steps in it.
%H G. C. Greubel, <a href="/A128726/b128726.txt">Table of n, a(n) for n = 0..1000</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*A128724(n,k).
%F G.f.: z(1-g+zg^2)/[(1-z-zg)(1+z-3zg)], where g=1+zg^2+z(g-1)=[1-z-sqrt(1-6z+5z^2)]/(2z).
%F a(n) ~ 5^(n-1/2)/(6*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014
%F Conjecture: +2*n*(n-3)^2*a(n) +(-13*n^3+84*n^2-167*n+100)*a(n-1) +(n-2)*(16*n^2-77*n+85)*a(n-2) -5*(n-3)*(n-2)^2*a(n-3)=0. - _R. J. Mathar_, Jun 17 2016
%e a(4)=7 because we have UDUUUDLL, UUUUDLLD, UUDUUDLL, UUUDUDLL, UUUUDDLL and UUUUDLLL (last path has two LL's).
%p g:=(1-z-sqrt(1-6*z+5*z^2))/(2*z): G:=z*(g-1-z*g^2)/((1-z-z*g)*(1+z-3*z*g)): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);
%t CoefficientList[Series[x*((1-x-Sqrt[1-6*x+5*x^2])/(2*x)-1-x*((1-x-Sqrt[1-6*x+5*x^2])/(2*x))^2)/((1-x-x*(1-x-Sqrt[1-6*x+5*x^2])/(2*x))*(1+x-3*x*(1-x-Sqrt[1-6*x+5*x^2])/(2*x))), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%Y Cf. A128724.
%K nonn
%O 0,5
%A _Emeric Deutsch_, Mar 31 2007
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