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%I #10 Jul 23 2017 12:16:48
%S 1,3,10,35,1,127,10,474,69,1810,406,3,7043,2193,49,27839,11252,496,
%T 111503,55858,3996,12,451640,271206,28120,270,1847032,1296584,181027,
%U 3575,7616692,6130552,1094856,36300,55,31637664,28753124,6325592,312832
%N Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k LDU's (n >= 0; 0 <= k <= floor((n-1)/3) for n >= 1).
%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.
%C Row n has ceiling(n/3) terms (n >= 1).
%C Row sums yield A002212.
%C T(n,0) = A128736(n).
%C Apparently, T(3k+1,k) = binomial(3k,k)/(2k+1) = A001764(k).
%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 Sum_{k>=0} k*T(n,k) = A128737(n).
%F G.f.: G = G(t,z) satisfies (t+1)zG^3 - (2 - 4z + 3tz)G^2 + 3(1 - 2z + tz)G - 1 + 2z - tz = 0.
%e T(7,2)=3 because we have UUUD(LDU)UUD(LDU)D, UUUUD(LDU)UD(LDU)D and UUUUUD(LDU)D(LDU)D (the LDU's are shown between parentheses).
%e Triangle starts:
%e 1;
%e 1;
%e 3;
%e 10;
%e 35, 1;
%e 127, 10;
%e 474, 69;
%e 1810, 406, 3;
%p eq:=(t+1)*z*G^3-(2-4*z+3*t*z)*G^2+3*(1-2*z+t*z)*G-1+2*z-t*z=0: G:=RootOf(eq,G): Gser:=simplify(series(G,z=0,18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser,z,n)) od: 1; for n from 1 to 15 do seq(coeff(P[n],t,j),j=0..floor((n-1)/3)) od; # yields sequence in triangular form
%Y Cf. A001764, A002212, A128736, A128737.
%K nonn,tabf
%O 0,2
%A _Emeric Deutsch_, Mar 31 2007
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