%I A102593
%S A102593 1,0,1,1,1,1,5,4,2,1,25,18,8,3,1,130,88,37,13,4,1,700,455,185,63,19,5,
               1,
%T A102593 3876,2448,973,325,97,26,6,1,21945,13566,5304,1748,518,140,34,7,1,
%U A102593 126500,76912,29697,9690,2856,775,193,43,8,1,740025,444015,169763,54967
%N A102593 Triangle read by rows: T(n,k) is the number of noncrossing trees with 
               n edges in which the maximum number of contiguous border edges starting 
               from the root in counterclockwise direction is equal to k.
%C A102593 Row n has n+1 terms. Row sums yield the ternary numbers (A001764). T(n,
               0)=A102893(n).
%D A102593 M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math.180, 
               1998, 301-313.
%D A102593 P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, 
               Discrete Math., 204, 1999, 203-229.
%F A102593 T(n, k)=(k+1)binomial(3n-2k, n-k)/(2n-k+1)-(k+2)binomial(3n-2k-2, n-k-1)/
               (2n-k) if n>1, 0<=k<=n; T(1, 1)=1; T(0, 0)=1; T(n, k)=0 if k>n. G.f.=G(t, 
               z)=g(1-zg)/(1-tzg), where g=1+zg^3 is the g.f. for the ternary numbers 
               (A001764).
%e A102593 T(2,0)=T(2,1)=T(2,2)=1 because in _\, /\ and /_ the maximum number of 
               contiguous border edges starting from the root in counterclockwise 
               direction is 0,1 and 2, respectively.
%e A102593 Triangle starts:
%e A102593 1;
%e A102593 0,1;
%e A102593 1,1,1;
%e A102593 5,4,2,1;
%e A102593 25,18,8,3,1;
%p A102593 T:=proc(n,k) if n=0 and k=0 then 1 elif n=1 and k=1 then 1 elif k<=n 
               then (k+1)*binomial(3*n-2*k,n-k)/(2*n-k+1)-(k+2)*binomial(3*n-2*k-2,
               n-k-1)/(2*n-k) else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) 
               od; # yields sequence in triangular form
%Y A102593 Cf. A001764, A102893.
%Y A102593 Sequence in context: A102220 A109430 A085917 this_sequence A090462 A081749 
               A074825
%Y A102593 Adjacent sequences: A102590 A102591 A102592 this_sequence A102594 A102595 
               A102596
%K A102593 nonn,tabl
%O A102593 0,7
%A A102593 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 22 2005