A129161 - OEIS

%I #7 Jul 23 2017 12:22:38

%S 1,1,2,1,5,4,1,11,16,8,1,23,53,44,16,1,47,165,186,112,32,1,95,494,725,

%T 568,272,64,1,191,1442,2707,2576,1600,640,128,1,383,4141,9813,11065,

%U 8184,4272,1472,256,1,767,11763,34827,45961,39026,24208,10976,3328,512

%N Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height k (1 <= k <= n).

%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 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,1) = 1;

%F T(n,2) = 3*2^(n-2) - 1 = A055010(n-1).

%F T(n,n) = 2^(n-1) = A000079(n-1).

%F Sum_{k=1..n} k*T(n,k) = A129162(n).

%F Column k has g.f. h[k]=H[k]-H[k-1], where H[k]=(1-z+zH[k-1])/(1-zH[k-1]), H[0]=1 (H[k] is the g.f. of paths of height at most k). For example, h[1]=z/(1-z); h[2]=z^2*(2-z)/[(1-z)(1-2z)]; h[3]=z^3*(2-z)^2/[(1-2z)(1-3z+z^2-z^3)].

%e T(3,2)=5 because we have UDUUDD, UDUUDL, UUDDUD, UUDUDD and UUDUDL.

%e Triangle starts:

%e   1;

%e   1,  2;

%e   1,  5,  4;

%e   1, 11, 16,  8;

%e   1, 23, 53, 44, 16;

%p H[0]:=1: for k from 1 to 11 do H[k]:=simplify((1+z*H[k-1]-z)/(1-z*H[k-1])) od: for k from 1 to 11 do h[k]:=factor(simplify(H[k]-H[k-1])) od: for k from 1 to 11 do hser[k]:=series(h[k],z=0,15) od: T:=(n,k)->coeff(hser[k],z,n): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

%Y Cf. A000079, A002212, A055010, A129162.

%K nonn,tabl

%O 1,3

%A _Emeric Deutsch_, Apr 03 2007