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A108263 Triangle read by rows: T(n,k) is the number of short bushes with n edges and k branchnodes (i.e., nodes of outdegree at least two). A short bush is an ordered tree with no nodes of outdegree 1. 9
1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 5, 0, 1, 9, 5, 0, 1, 14, 21, 0, 1, 20, 56, 14, 0, 1, 27, 120, 84, 0, 1, 35, 225, 300, 42, 0, 1, 44, 385, 825, 330, 0, 1, 54, 616, 1925, 1485, 132, 0, 1, 65, 936, 4004, 5005, 1287, 0, 1, 77, 1365, 7644, 14014, 7007, 429, 0, 1, 90, 1925, 13650, 34398 (list; graph; refs; listen; history; text; internal format)
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.

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

Cf. A005043, A033275, A033276.

Columns: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279; A005043 (row sums), A033282, A086810

Sequence in context: A112899 A212808 A209687 * A244523 A325304 A134433

Adjacent sequences:  A108260 A108261 A108262 * A108264 A108265 A108266

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, May 29 2005

STATUS

approved

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Last modified November 18 22:26 EST 2019. Contains 329306 sequences. (Running on oeis4.)