OFFSET
0,9
COMMENTS
Row n has 1+floor(n/2) terms. Row sums are the Riordan numbers (A005043). Column 3 yields A033275; column 4 yields A033276.
Related to the number of certain non-crossing partitions for the root system A_n. Cf. p. 12, Athanasiadis and Savvidou. Diagonals are A033282/A086810. Also see A132081 and A100754.- Tom Copeland, Oct 19 2014
LINKS
Indranil Ghosh, Rows 0..100, flattened
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362, 2012
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 10.
FORMULA
G.f. G=G(t, z) satisfies z*(1+t*z)*G^2 - (1+z)*G + 1 = 0.
T(n, k) = A086810(n-k, k). - Philippe Deléham, May 30 2005
EXAMPLE
T(6,3)=5 because the only short bushes with 6 edges and 3 branchnodes are the five full binary trees with 6 edges.
Triangle begins:
1;
0;
0,1;
0,1;
0,1,2;
0,1,5;
0,1,9,5
MAPLE
G:=(1+z-sqrt((1-z)^2-4*t*z^2))/2/z/(1+t*z): Gser:=simplify(series(G, z=0, 18)): P[0]:=1: for n from 1 to 16 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 16 do seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)) od; # yields sequence in triangular form
A108263 := (n, k) -> binomial(n-k-1, n-2*k)*binomial(n, k)/(n-k+1);
seq(print(seq(A108263(n, k), k=0..ceil((n-1)/2))), n=0..8); # Peter Luschny, Sep 25 2014
MATHEMATICA
T[n_, k_]:=Binomial[n-k-1, n-2k]*Binomial[n, k]/(n-k+1); Flatten[Table[T[n, k], {n, 0, 11}, {k, 0, Ceiling[(n-1)/2]}]] (* Indranil Ghosh, Feb 20 2017 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, May 29 2005
STATUS
approved