Search: id:A101449
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%I A101449
%S A101449 1,1,2,4,4,4,11,24,12,8,41,88,96,32,16,146,410,440,320,80,32,564,1752,
%T A101449 2460,1760,960,192,64,2199,7896,12264,11480,6160,2688,448,128,8835,
%U A101449 35184,63168,65408,45920,19712,7168,1024,256,35989,159030,316656,379008
%N A101449 Triangle read by rows: T(n,k) is number of noncrossing trees with n edges 
               and having k nonroot nodes of degree 1.
%C A101449 Row n contains n terms. Row sums yield the ternary numbers (A001764). 
               Column 0 is A030981. T(n,k)=2^k*binomial(n-1,k)*A030981(n-k). The 
               average number of nonroot nodes of degree 1 over all noncrossing 
               trees with n edges is 4n(n-1)(2n+1)/[3(3n-1)(3n-2)] ~ 8n/27.
%D A101449 P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, 
               Discrete Math., 204, 1999, 203-229.
%D A101449 M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math. 
               180, 1998, 301-313.
%F A101449 T(n, k)=[2^k/(n-k)]*binomial(n-1, k)*sum((-1)^(n-k-i)*2^(n-k-i)*binomial(n-k, 
               i)*binomial(3i, i-1), i=1..n-k) (0<=k<n).
%e A101449 T(2,0)=1 (/\); T(2,1)=2 (/_, _\ ).
%e A101449 Triangle begins:
%e A101449 1;
%e A101449 1,2;
%e A101449 4,4,4;
%e A101449 11,24,12,8;
%e A101449 41,88,96,32,16;
%p A101449 T:=proc(n,k) if k<n then 2^k*binomial(n-1,k)*sum((-1)^(n-k-i)*2^(n-k-i)*binomial(n-k,
               i)*binomial(3*i,i-1),i=1..n-k)/(n-k) else 0 fi end: for n from 1 
               to 10 do seq(T(n,k),k=0..n-1) od; #yields sequence in triangular 
               form
%Y A101449 Cf. A001764, A030981.
%Y A101449 Sequence in context: A021413 A082855 A107058 this_sequence A134188 A140295 
               A070529
%Y A101449 Adjacent sequences: A101446 A101447 A101448 this_sequence A101450 A101451 
               A101452
%K A101449 nonn,tabl
%O A101449 1,3
%A A101449 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 17 2005

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