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%I #14 Oct 27 2015 05:25:02
%S 1,1,1,1,4,0,1,11,2,0,1,33,6,2,0,1,105,17,7,2,0,1,343,56,19,8,2,0,1,
%T 1148,185,64,21,9,2,0,1,3916,624,214,72,23,10,2,0,1,13563,2144,726,
%U 244,80,25,11,2,0,1,47571,7468,2510,832,275,88,27,12,2,0,1,168625,26317
%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k UDUD's starting at level 0; here U=(1,1), D=(1,-1) (0<=k<=n-1).
%C Row 0 has one entry; row n has n entries (n>=1). Row sums yield the Catalan numbers (A000108). Column 0 yields A127154. The reference does not list the 0's (p. 2920, lines 3,4).
%H A. Sapounakis, I. Tasoulas and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.005">Counting strings in Dyck paths</a>, Discrete Math., 307 (2007), 2909-2924.
%F G.f.: (1+z-t*z)/(1+z-t*z+z^2-t*z^2-z*C(1+z-t*z)), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan g.f. (see A000108).
%F Or, g.f.: (1+(1-t)*z)*C/(1+(1-t)*z*(1+z*C)).
%e T(4,1)=2 because we have UDUDUUDD and UUDDUDUD; T(4,3)=1 because we have UDUDUDUD.
%e Triangle starts:
%e 1;
%e 1;
%e 1,1;
%e 4,0,1;
%e 11,2,0,1;
%e 33,6,2,0,1;
%e 105,17,7,2,0,1;
%p G:=(1+z-t*z)/(1+z-t*z+z^2-t*z^2-z*C*(1+z-t*z)): C:=(1-sqrt(1-4*z))/2/z: Gser:=simplify(series(G,z=0,16)): for n from 0 to 13 do P[n]:=sort(coeff(Gser,z,n)) od: 1; for n from 1 to 13 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form
%Y Cf. A000108, A127154.
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Feb 27 2007
%E Edited by _N. J. A. Sloane_, May 16 2008 at the suggestion of _R. J. Mathar_
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