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A128727 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k DDU and LDU's.

%I A128727 #21 Oct 30 2018 09:40:38
%S A128727 1,1,3,9,1,27,9,81,54,2,243,270,30,729,1215,270,5,2187,5103,1890,105,
%T A128727 6561,20412,11340,1260,14,19683,78732,61236,11340,378,59049,295245,
%U A128727 306180,85050,5670,42,177147,1082565,1443420,561330,62370,1386,531441
%N A128727 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k DDU and LDU's.
%C A128727 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.
%C A128727 Row n has ceiling(n/2) terms (n >= 1).
%C A128727 Row sums yield A002212.
%C A128727 Apparently a(n) = A126177(n-1). - _Georg Fischer_, Oct 28 2018
%H A128727 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 A128727 T(n,0) = 3^(n-1).
%F A128727 T(2k+1,k) = binomial(2k,k)/(k+1) (the Catalan numbers, A000108).
%F A128727 T(2k,k-1) = 3binomial(2k-1,k) = A003409(k).
%F A128727 Sum_{k>=0} k*T(n,k) = A026377(n-1).
%F A128727 T(n,k) = (1/n)*3^(n-1-2k)*binomial(n,k)*binomial(n-k,k+1).
%F A128727 G.f.: G = G(t,z) satisfies tzG^2 - (1 - 3z + 2tz)G + 1 - 2z + tz = 0.
%e A128727 T(5,2)=2 because we have UU(DDU)U(DDU)D and UUU(DDU)(DDU)D (the 2 subwords are shown between parentheses).
%e A128727 Triangle starts:
%e A128727     1;
%e A128727     1;
%e A128727     3;
%e A128727     9,    1;
%e A128727    27,    9;
%e A128727    81,   54,    2;
%e A128727   243,  270,   30;
%e A128727   729, 1215,  270,    5;
%p A128727 T:=(n,k)->3^(n-1-2*k)*binomial(n,k)*binomial(n-k,k+1)/n: 1; for n from 1 to 13 do seq(T(n,k),k=0..floor((n-1)/2)) od; # yields sequence in triangular form
%Y A128727 Cf. A000108, A002212, A003409, A026377, A126177.
%K A128727 nonn,tabf
%O A128727 0,3
%A A128727 _Emeric Deutsch_, Mar 31 2007