%I A068218
%S A068218 1,2,2,2,16,2,4,84,84,4,10,400,1056,400,10,28,1820,9184,9184,1820,28,84,
%T A068218 8064,66276,126720,66276,8064,84,264,35112,426888,1329768,1329768,
%U A068218 426888,35112,264,858,151008,2546544,11737440,19123776,11737440
%N A068218 Triangle of numbers of square lattice walks that start and end at origin 
               after 2k steps and contain exactly r steps to the east, not touching 
               origin at intermediate stages.
%C A068218 The given recurrences do not provide a means to calculate T(2r,r). But 
               T(2r,r) is computable by the formula relating T(k,r) to A069466(k,
               r).
%F A068218 T(k, r) = 2*(2k-3)/(k-2r) * ( T(k-1, r) - T(k-1, r-1) ), for k > 2r. 
               T(1, 0)=2, T(1, 1)=2 Sum[T(k, r), r=0, ..., k] = A054474(k) T(k, 
               r)=A069466(k, r) - Sum[ Sum[ T(i, j)*A069466(k-i, r-j), j=0...r], 
               i=1, k-1]
%e A068218 T(3,1)=84 because there are 84 distinct lattice walks of length 2*3=6 
               starting and ending at the origin and containing exactly 1 step to 
               the east and not touching origin at intermediate steps. Let E, W, 
               S, N denote the 4 possible directions, then NNEWSS and NWSSNE are 
               examples of such walks.
%Y A068218 T(k, 0) = A002420(k) = A069466(k)/(2k-1).
%Y A068218 Cf. A054474 (row sums).
%Y A068218 Sequence in context: A074052 A129409 A025521 this_sequence A098919 A161748 
               A079007
%Y A068218 Adjacent sequences: A068215 A068216 A068217 this_sequence A068219 A068220 
               A068221
%K A068218 easy,nice,nonn,tabl
%O A068218 0,2
%A A068218 Martin Wohlgemuth (mail(AT)matroid.com), Mar 24 2002