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A101449 Triangle read by rows: T(n,k) is number of noncrossing trees with n edges and having k nonroot nodes of degree 1. 3

%I #19 Jul 06 2018 03:02:21

%S 1,1,2,4,4,4,11,24,12,8,41,88,96,32,16,146,410,440,320,80,32,564,1752,

%T 2460,1760,960,192,64,2199,7896,12264,11480,6160,2688,448,128,8835,

%U 35184,63168,65408,45920,19712,7168,1024,256,35989,159030,316656,379008

%N Triangle read by rows: T(n,k) is number of noncrossing trees with n edges and having k nonroot nodes of degree 1.

%C Row n contains n terms.

%C Row sums yield the ternary numbers (A001764).

%C 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.

%H Andrew Howroyd, <a href="/A101449/b101449.txt">Table of n, a(n) for n = 1..1275</a>

%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^k/(n-k)]*binomial(n-1, k)*Sum_{i=1..n-k} (-1)^(n-k-i)*2^(n-k-i)*binomial(n-k, i)*binomial(3i, i-1), 0 <= k < n).

%F T(n,k) = 2^k*binomial(n-1,k)*A030981(n-k).

%e T(2,0)=1 (/\); T(2,1)=2 (/_, _\ ).

%e Triangle begins:

%e 1;

%e 1, 2;

%e 4, 4, 4;

%e 11, 24, 12, 8;

%e 41, 88, 96, 32, 16;

%p 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

%t T[n_, k_] := If[k<n, 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), 0];

%t Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Jul 06 2018, from Maple *)

%o (PARI) T(n,k)={if(k<n, 2^k*binomial(n-1, k)*sum(i=1, n-k, (-1)^(n-k-i)*2^(n-k-i)*binomial(n-k, i)*binomial(3*i, i-1))/(n-k))}

%o for(n=1, 10, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 06 2017

%Y Cf. A001764, A030981 (column 0).

%K nonn,tabl

%O 1,3

%A _Emeric Deutsch_, Jan 17 2005

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)