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%I #12 Jul 24 2022 10:37:31
%S 0,0,0,0,1,10,69,412,2291,12244,63886,328256,1669363,8429384,42349096,
%T 211982828,1058244079,5272285552,26227527576,130323237088,
%U 647013004499,3210128312122,15919166804461,78915323039268,391100149306301
%N Number of LDU'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 the path is defined to be the number of its steps.
%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} k*A128735(n,k).
%F G.f.: z(g-1)^3/(4g - 2zg - 6zg^2 - 3 + 3*z), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
%F Conjecture D-finite with recurrence: -10*(n+1)*(n-4)*a(n) +(73*n^2-273*n+140)*a(n-1) +(-132*n^2+641*n-734) *a(n-2) +(n-3)*(89*n-269)*a(n-3) -20*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jun 17 2016
%e a(4)=1 because among the 36 (=A002212(4)) skew Dyck paths of semilength 4 only UUUDLDUD has a LDU.
%p g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: ser:=series(z*(g-1)^3/(4*g-2*z*g-6*z*g^2-3+3*z),z=0,30): seq(coeff(ser,z,n),n=0..27);
%Y Cf. A128735.
%K nonn
%O 0,6
%A _Emeric Deutsch_, Mar 31 2007