%I A101372
%S A101372 1,2,2,7,10,4,30,50,32,8,143,260,208,88,16,728,1400,1280,704,224,32,
%T A101372 3876,7752,7752,5016,2128,544,64,21318,43890,46816,33880,17248,5984,
%U A101372 1280,128,120175,253000,283360,222640,128800,54400,16000,2944,256
%N A101372 Triangle read by rows: T(n,k) is number of leaves at level k in all noncrossing 
               rooted trees on n+1 nodes.
%C A101372 Row n has n terms. Row sums yield A045721. Column 1 is A006013.
%D A101372 P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, 
               Discrete Math. 204 (1999), 203-229.
%D A101372 M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 
               180, 301-313, 1998.
%F A101372 T(n, k)=2^(k-1)*[(3k-1)/(2n+k-1)]binomial(3n-2, n-k) (1<=k<=n). G.f.=tzg^2/
               (1-2tzg^3), where g=1+zg^3 is the g.f. of the ternary numbers (A001764).
%e A101372 Triangle begins:
%e A101372 1;
%e A101372 2,2;
%e A101372 7,10,4;
%e A101372 30,50,32,8;
%e A101372 143,260,208,88,16;
%p A101372 T:=(n,k)->2^(k-1)*(3*k-1)*binomial(3*n-2,n-k)/(2*n+k-1): for n from 1 
               to 10 do seq(T(n,k),k=1..n) od; # yields triangle in triangular form
%Y A101372 Cf. A045721, A006013.
%Y A101372 Sequence in context: A070910 A107386 A095021 this_sequence A133374 A054226 
               A000024
%Y A101372 Adjacent sequences: A101369 A101370 A101371 this_sequence A101373 A101374 
               A101375
%K A101372 nonn,tabl
%O A101372 1,2
%A A101372 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 14 2005