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%I #16 Jul 29 2018 19:54:24
%S 1,2,1,4,4,4,8,12,24,11,16,32,96,88,41,32,80,320,440,410,146,64,192,
%T 960,1760,2460,1752,564,128,448,2688,6160,11480,12264,7896,2199,256,
%U 1024,7168,19712,45920,65408,63168,35184,8835,512,2304,18432,59136,165312,294336,379008,316656,159030,35989
%N Triangle read by rows: T(n,k) is number of noncrossing trees with n edges and having k branches.
%C Mirror image of A101449.
%C T(n,k) = 2^(n-k)*binomial(n-1,k-1)*A030981(k).
%C Row sums yield the ternary numbers (A001764).
%C T(n,n) = A030981(n).
%C 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).
%H 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 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 T(n, k) = [2^(n-k)/k]binomial(n-1, k-1)*Sum_{i=1..k} (-2)^(k-i)*binomial(k, i)*binomial(3i, i-1).
%F 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 T(2,1)=2 because we have /_ and _\; T(2,2)=1 because we have /\
%e Triangle begins:
%e 1;
%e 2, 1;
%e 4, 4, 4;
%e 8, 12, 24, 11;
%e 16, 32, 96, 88, 41;
%e ...
%p 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
%t 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}];
%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 29 2018 *)
%Y Cf. A001764, A030981, A045738.
%K nonn,tabl
%O 1,2
%A _Emeric Deutsch_, Jan 19 2005
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