A128733 - OEIS

%I #8 Jul 23 2017 12:16:40

%S 1,3,9,1,28,8,90,46,1,297,231,15,1001,1079,138,1,3432,4823,1006,24,

%T 11934,20944,6388,320,1,41990,89148,37026,3170,35,149226,374034,

%U 201210,26130,635,1,534888,1552661,1042492,189959,8170,48,1931540,6393310

%N Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k LD's (n>=0; 0<=k<=floor((n-1)/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 a path is defined to be the number of its steps.

%C Row n has ceiling(n/2) terms (n >= 1).

%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) = 3(2n)!/((n+2)!(n-1)!) = A000245(n) (n >= 1).

%F Sum_{k=0..floor((n-1)/2)} k*T(n,k) = A128734(n).

%F G.f.: G = G(t,z) satisfies tz^2*G^3 - (t-1)z^2*G^2 - (1 - 3z + 2z^2)G + (1-z)^2 = 0.

%e T(3,1)=1 because we have UUUDLD.

%e Triangle starts:

%e     1;

%e     1;

%e     3;

%e     9,   1;

%e    28,   8;

%e    90,  46,   1;

%e   297, 231,  15;

%p eq:=t*z^2*G^3-(t-1)*z^2*G^2-(1-3*z+2*z^2)*G+(1-z)^2=0: G:=RootOf(eq,G): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor((n-1)/2)) od; # yields sequence in triangular form

%Y Cf. A000245, A002212, A128734.

%K nonn,tabf

%O 0,2

%A _Emeric Deutsch_, Mar 31 2007