%I A098978
%S A098978 1,1,1,1,2,3,5,8,1,13,23,6,35,69,27,1,97,212,110,10,275,662,426,66,1,
%T A098978 794,2091,1602,360,15,2327,6661,5912,1760,135,1
%N A098978 Triangle read by rows: T(n,k) is number of Dyck n-paths with k UUDDs, 
               0 <= k <= n/2.
%C A098978 T(n,k) is the number of Lukasiewicz paths of length n having k peaks. 
               A Lukasiewicz path of length n is a path in the first quadrant from 
               (0,0) to (n,0) using rise steps (1,k) for any positive integer k, 
               level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative 
               Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 
               223, Exercise 6.19w; the integers are the slopes of the steps). Example: 
               T(3,1)=3 because we have HUD, UDH and U(2)DD, where H=(1,0), U(1,
               1), U(2)=(1,2) and D=(1,-1). R. P. Stanley, Enumerative Combinatorics, 
               Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 
               6.19w (the integers are the slopes of the steps). - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Jan 06 2005
%F A098978 G.f. (1 + z^2 - t*z^2 - (-4*z + (-1 - z^2 + t*z^2)^2)^(1/2))/(2*z) = 
               Sum_{n>=0, 0<=k<=n/2}T(n, k)z^n*t^k and it satisfies G = 1 + G^2*z 
               + G*(-z^2 + t*z^2).
%F A098978 T(n,k) = Sum((-1)^j * binomial(n-(j+k),j+k) * binomial(2n - 3(j+k), n-(j+k)-1) 
               * binomial(j+k,k)/(n-(j+k)), j=0..[n/2]-k). - I. Tasoulas (jtas(AT)unipi.gr), 
               Feb 19 2006
%e A098978 Table begins
%e A098978 \ k 0, 1, 2, ...
%e A098978 n
%e A098978 0 | 1
%e A098978 1 | 1
%e A098978 2 | 1, 1
%e A098978 3 | 2, 3
%e A098978 4 | 5, 8, 1
%e A098978 5 | 13, 23, 6
%e A098978 6 | 35, 69, 27, 1
%e A098978 7 | 97, 212, 110, 10
%e A098978 8 |275, 662, 426, 66, 1
%e A098978 T(3,1)=3 because each of UUUDDD, UDUUDD, UUDDUD has one UUDD.
%Y A098978 column k=0 is A025242 (apart from first term).
%Y A098978 Sequence in context: A093092 A031111 A089911 this_sequence A111301 A096320 
               A105955
%Y A098978 Adjacent sequences: A098975 A098976 A098977 this_sequence A098979 A098980 
               A098981
%K A098978 nonn
%O A098978 0,5
%A A098978 David Callan (callan(AT)stat.wisc.edu), Oct 24 2004