%I #15 Sep 12 2024 19:33:46

%S 1,3,8,4,20,30,5,48,144,75,6,112,560,595,154,7,256,1920,3440,1848,280,

%T 8,576,6048,16380,14994,4788,468,9,1280,17920,68320,95200,52290,10920,

%U 735,10,2816,50688,258720,510048,429198,155496,22638,1100,11

%N Triangle T(n,k) (n >= 2, 2 <= k <= n-1 if n > 2) giving number of non-crossing trees with n nodes and k endpoints.

%C For n > 2 the n-th row has n-2 terms.

%C The difference between this sequence and A091320 is that this sequence considers the degrees of all vertices whereas A091320 ignores the degree of the root vertex. - _Andrew Howroyd_, Nov 06 2017

%H E. Deutsch and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00366-1">Statistics on non-crossing trees</a>, Discrete Math., 254 (2002), 75-87.

%F T(n, k) = U(n, k-1) - U(n, k) + binomial(n-1, k)*Sum_{j=0..k-1} binomial(n-1, j)*binomial(n-k-1, k-1-j)*2^(n-2*k+j)/(n-1) where U(n,k) = 2*binomial(n-2, k)*Sum_{j=0..k-1} binomial(n-1, j)*binomial(n-k-2, k-1-j)*2^(n-1-2*k+j)/(n-2) for 2 < k < n. - _Andrew Howroyd_, Nov 06 2017

%e Triangle begins:

%e 1;

%e 3;

%e 8, 4;

%e 20, 30, 5;

%e 48, 144, 75, 6;

%e ...

%t U[n_, k_] := 2 Binomial[n - 2, k] Sum[Binomial[n - 1, j] Binomial[n - k - 2, k - 1 - j] 2^(n - 1 - 2k + j), {j, 0, k - 1}]/(n - 2);

%t W[n_, k_] := Binomial[n - 1, k] Sum[Binomial[n - 1, j] Binomial[n - k - 1, k - 1 - j] 2^(n - 2k + j), {j, 0, k - 1}]/(n - 1);

%t T[n_, k_] := If[n < 3, n == 2 && k == 2, If[1 < k && k < n, U[n, k - 1] - U[n, k] + W[n, k]]];

%t Table[T[n, k] /. True -> 1, {n, 2, 10}, {k, 2, n-Boole[n>2]}] // Flatten (* _Jean-François Alcover_, Sep 06 2019, from PARI *)

%o (PARI)

%o U(n,k) = 2*binomial(n-2,k)*sum(j=0,k-1,binomial(n-1,j)*binomial(n-k-2,k-1-j)*2^(n-1-2*k+j))/(n-2);

%o W(n,k) = binomial(n-1, k)*sum(j=0,k-1,binomial(n-1,j)*binomial(n-k-1,k-1-j)*2^(n-2*k+j))/(n-1);

%o T(n,k) = if(n<3, n==2&&k==2, if(1<k&&k<n, U(n,k-1)-U(n,k)+W(n,k)));

%o for(n=2, 10, for(k=2, n-(n>2), print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 06 2017

%Y Column k=2 gives A001792, row sums are A001764.

%Y Cf. A091320.

%K nonn,tabf

%O 2,2

%A _N. J. A. Sloane_, Jul 06 2002

%E Offset corrected by _Andrew Howroyd_, Nov 06 2017

%E More terms from _Sean A. Irvine_, Sep 12 2024