A128719 - OEIS

%I #10 Jul 23 2017 03:03:00

%S 1,1,3,6,4,16,12,8,40,53,28,16,109,176,162,64,32,297,625,633,456,144,

%T 64,836,2084,2677,2024,1216,320,128,2377,7016,10257,9849,6008,3120,

%U 704,256,6869,23218,39378,42222,32930,16928,7776,1536,512,20042,76811,146191

%N 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 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 n-1 terms (n >= 2).

%C Row sums yield A002212.

%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 T(n,0) = A128720(n).

%F Sum_{k=0..n-2} k*T(n,k) = A128721(n) for n >= 2.

%F G.f.: G = G(t,z) satisfies z(t + z - tz)G^2 - (1 - z - z^2 + tz^2)G + 1 - tz = 0.

%e T(3,1)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.

%e Triangle starts:

%e    1;

%e    1;

%e    3;

%e    6,  4;

%e   16, 12,  8;

%e   40, 53, 28, 16;

%p 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 Cf. A002212, A128720, A128721.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Mar 30 2007