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A101372 Triangle read by rows: T(n,k) is number of leaves at level k in all noncrossing rooted trees on n+1 nodes.

%I A101372 #10 Oct 26 2015 12:57:56
%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.
%H A101372 P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of non-crossing configurations</a>, Discrete Math., 204, 203-229, 1999.
%H A101372 M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00121-0">Enumeration of noncrossing trees on a circle</a>, 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).
%F A101372 G.f.: t*z*g^2/(1-2*t*z*g^3), where g = 1 + z*g^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;
%e A101372 ...
%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
%t A101372 Flatten[Table[2^(k-1) ((3k-1)/(2n+k-1))Binomial[3n-2,n-k],{n,10},{k,n}]] (* _Harvey P. Dale_, Feb 10 2015 *)
%Y A101372 Cf. A045721, A006013.
%K A101372 nonn,tabl
%O A101372 1,2
%A A101372 _Emeric Deutsch_, Jan 14 2005