OFFSET
3,2
COMMENTS
Row sums give the little Schroeder numbers (A001003). Column 3 (first column, corresponding to k=3) gives A010683.
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.
LINKS
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.
FORMULA
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).
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).
EXAMPLE
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.
Triangle starts:
1;
2,1;
7,3,1;
28,12,4,1;
121,52,18,5,1;
...
MAPLE
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);
T_row := proc(n) local c, f, s;
c := N -> hypergeom([1-N, N+2], [2], -1);
f := n -> 1+add(simplify(c(i))*x^i, i=1..n):
s := j -> coeff(series(f(j)^2/(1-x*t*f(j)), x, j+1), x, j):
seq(coeff(s(n), t, j), j=0..n) end:
seq(T_row(n), n=0..9); # Peter Luschny, Oct 30 2015
MATHEMATICA
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}];
Table[T[n, k], {n, 3, 13}, {k, 3, n}] // Flatten (* Jean-François Alcover, Nov 24 2017 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Mar 01 2004
STATUS
approved