%I A101452
%S A101452 1,2,1,4,4,4,8,12,24,11,16,32,96,88,41,32,80,320,440,410,146,64,192,960,
%T A101452 1760,2460,1752,564,128,448,2688,6160,11480,12264,7896,2199,256,1024,
%U A101452 7168,19712,45920,65408,63168,35184,8835,512,2304,18432,59136,165312
%N A101452 Triangle read by rows: T(n,k) is number of noncrossing trees with n edges 
               and having k branches.
%C A101452 Mirror image of A101449. T(n,k)=2^(n-k)*binomial(n-1,k-1)*A030981(k). 
               Row n contains n terms. Row sums yield the ternary numbers (A001764). 
               T(n,n)=A030981(n) The average number of branchnodes over all noncrossing 
               trees with n edges is n(n-1)(19n^2-23n+10)/[3(3n-1)(3n-2)] ~ 19n/
               27 (see A045738).
%D A101452 P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, 
               Discrete Math., 204, 1999, 203-229.
%D A101452 M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math. 
               180, 1998, 301-313.
%F A101452 T(n, k)=[2^(n-k)/k]binomial(n-1, k-1)*sum((-2)^(k-i)*binomial(k, i)*binomial(3i, 
               i-1), i=1..k). G.f.=G(t, z)=1/(1-F), where F satisfies F=z[t+2tF^2/
               (1-F)+tF^2/(1-F)^2+2F].
%e A101452 T(2,1)=2 because we have /_ and _\; T(2,2)=1 because we have /\
%e A101452 Triangle begins:
%e A101452 1;
%e A101452 2,1;
%e A101452 4,4,4;
%e A101452 8,12,24,11;
%e A101452 16,32,96,88,41;
%p A101452 T:=(n,k)->(2^(n-k)/k)*binomial(n-1,k-1)*sum((-2)^(k-i)*binomial(k,i)*binomial(3*i,
               i-1),i=1..k):for n from 1 to 10 do seq(T(n,k),k=1..n) od; #yields 
               sequence in triangular form
%Y A101452 Cf. A001764, A030981, A045738.
%Y A101452 Sequence in context: A111975 A117250 A136692 this_sequence A019963 A165417 
               A108755
%Y A101452 Adjacent sequences: A101449 A101450 A101451 this_sequence A101453 A101454 
               A101455
%K A101452 nonn,tabl
%O A101452 1,2
%A A101452 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 19 2005