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A128753 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDUDU's (n >= 0; 0 <= k <= n-2 for n >= 2).

%I A128753 #11 Jul 23 2017 12:17:53
%S A128753 1,1,3,9,1,31,4,1,113,19,4,1,431,86,21,4,1,1697,393,101,23,4,1,6847,
%T A128753 1800,492,116,25,4,1,28161,8279,2388,596,131,27,4,1,117631,38218,
%U A128753 11603,3032,705,146,29,4,1,497665,177013,56407,15403,3732,819,161,31,4,1
%N A128753 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDUDU's (n >= 0; 0 <= k <= n-2 for n >= 2).
%C A128753 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 its steps.
%C A128753 Rows 0 and 1 have one term each; row n has n-1 terms (n >= 2).
%C A128753 Row sums yield A002212.
%H A128753 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 A128753 T(n,0) = A052709(n+1).
%F A128753 Sum_{k=0..n-2} k*T(n,k) = A026376(n-2).
%F A128753 G.f.: G = G(t,z) satisfies z(1 + z - tz)G^2 - (1 - tz)G + 1 - tz = 0. G = C((1+z-tz)/(1-tz)), where C(z) = (1 - sqrt(1 - 4z))/(2z) is the Catalan function.
%e A128753 T(4,1)=4 because we have (UDUDU)UDD, (UDUDU)UDL, U(UDUDU)DD and U(UDUDU)DL (the subwords UDUDU are shown between parentheses).
%e A128753 Triangle starts
%e A128753     1;
%e A128753     1;
%e A128753     3;
%e A128753     9,  1;
%e A128753    31,  4,  1;
%e A128753   113, 19,  4,  1;
%p A128753 C:=z->(1-sqrt(1-4*z))/2/z: G:=C(z*(1+z-t*z)/(1-t*z)): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: 1; 1; for n from 2 to 12 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form
%Y A128753 Cf. A002212, A052709, A026376.
%K A128753 nonn,tabf
%O A128753 0,3
%A A128753 _Emeric Deutsch_, Apr 01 2007