%I #23 Nov 24 2017 04:31:58
%S 1,2,1,7,3,1,28,12,4,1,121,52,18,5,1,550,237,84,25,6,1,2591,1119,403,
%T 125,33,7,1,12536,5424,1976,630,176,42,8,1,61921,26832,9860,3206,930,
%U 238,52,9,1,310954,134913,49912,16470,4908,1316,312,63,10,1
%N Triangle read by rows: T(n,k) is the number of dissections of a convex n-gon by nonintersecting diagonals, having a k-gon over a fixed edge (base).
%C Row sums give the little Schroeder numbers (A001003). Column 3 (first column, corresponding to k=3) gives A010683.
%C Number of short bushes (i.e. ordered trees with no vertices of outdegree 1) with n-1 leaves and having root of degree k-1. Example: T(5,3)=7 because, in addition to the five binary trees with 6 edges we have (i) two edges rb, rc hanging from the root r with three edges hanging from vertex b and (ii) two edges rb, rc hanging from the root r with three edges hanging from vertex c.
%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 J.-C. Novelli and J.-Y. Thibon, <a href="http://arXiv.org/abs/math.CO/0512570">Noncommutative Symmetric Functions and Lagrange Inversion</a>, arXiv:math/0512570 [math.CO], 2005-2006.
%F T(n, k) = [(k-1)/(n-k)]*sum(2^j*binomial(n-2, n-k-1-j)*binomial(n-k, j), j=0..n-k-1).
%F G.f.: t^3*z^3*S^2/(1-t*z*S), where S = (1+z-sqrt(1-6*z+z^2))/(4*z) is the g.f. of the little Schroeder numbers (A001003).
%e T(5,4)=3 because the dissections of the pentagon ABCDEA that have a quadrilateral over the base AB are obtained by the diagonals (i) CE, (ii) AD and (iii) BD, respectively.
%e Triangle starts:
%e 1;
%e 2,1;
%e 7,3,1;
%e 28,12,4,1;
%e 121,52,18,5,1;
%e ...
%p a := proc(n,k) if k=0 or k=1 or k=2 then 0 elif k=n then 1 elif k<n then (k-1)*sum(2^j*binomial(n-2,n-k-1-j)*binomial(n-k,j),j=0..n-k-1)/(n-k) else 0 fi end:seq(seq(a(n,k),k=3..n),n=3..13);
%p T_row := proc(n) local c,f,s;
%p c := N -> hypergeom([1-N, N+2], [2], -1);
%p f := n -> 1+add(simplify(c(i))*x^i,i=1..n):
%p s := j -> coeff(series(f(j)^2/(1-x*t*f(j)),x,j+1),x,j):
%p seq(coeff(s(n),t,j),j=0..n) end:
%p seq(T_row(n), n=0..9); # _Peter Luschny_, Oct 30 2015
%t T[n_, n_] = 1; T[n_, k_] := (k - 1)/(n - k)*Sum[2^j*Binomial[n - 2, n - k - 1 - j]*Binomial[n - k, j], {j, 0, n - k - 1}];
%t Table[T[n, k], {n, 3, 13}, {k, 3, n}] // Flatten (* _Jean-François Alcover_, Nov 24 2017 *)
%Y Cf. A001003, A010683.
%K nonn,tabl
%O 3,2
%A _Emeric Deutsch_, Mar 01 2004