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A128719 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UUU's (triplerises) (n >= 0; 0 <= k <= n-2 for n >= 2).

%I A128719 #10 Jul 23 2017 03:03:00
%S A128719 1,1,3,6,4,16,12,8,40,53,28,16,109,176,162,64,32,297,625,633,456,144,
%T A128719 64,836,2084,2677,2024,1216,320,128,2377,7016,10257,9849,6008,3120,
%U A128719 704,256,6869,23218,39378,42222,32930,16928,7776,1536,512,20042,76811,146191
%N A128719 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UUU's (triplerises) (n >= 0; 0 <= k <= n-2 for n >= 2).
%C A128719 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 A128719 Row n has n-1 terms (n >= 2).
%C A128719 Row sums yield A002212.
%H A128719 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 A128719 T(n,0) = A128720(n).
%F A128719 Sum_{k=0..n-2} k*T(n,k) = A128721(n) for n >= 2.
%F A128719 G.f.: G = G(t,z) satisfies z(t + z - tz)G^2 - (1 - z - z^2 + tz^2)G + 1 - tz = 0.
%e A128719 T(3,1)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
%e A128719 Triangle starts:
%e A128719    1;
%e A128719    1;
%e A128719    3;
%e A128719    6,  4;
%e A128719   16, 12,  8;
%e A128719   40, 53, 28, 16;
%p A128719 eq:=z*(t+z-t*z)*G^2-(1-z-z^2+t*z^2)*G+1-t*z=0: G:=RootOf(eq,G): Gser:=simplify(series(G,z=0,14)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: 1; 1; for n from 2 to 11 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form
%Y A128719 Cf. A002212, A128720, A128721.
%K A128719 nonn,tabf
%O A128719 0,3
%A A128719 _Emeric Deutsch_, Mar 30 2007