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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.

%I A108263 #31 May 31 2023 09:50:41
%S A108263 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,
%T A108263 0,1,35,225,300,42,0,1,44,385,825,330,0,1,54,616,1925,1485,132,0,1,65,
%U A108263 936,4004,5005,1287,0,1,77,1365,7644,14014,7007,429,0,1,90,1925,13650,34398
%N 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.
%C A108263 Row n has 1+floor(n/2) terms. Row sums are the Riordan numbers (A005043). Column 3 yields A033275; column 4 yields A033276.
%C A108263 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
%H A108263 Indranil Ghosh, <a href="/A108263/b108263.txt">Rows 0..100, flattened</a>
%H A108263 C. Athanasiadis and C. Savvidou, <a href="http://arxiv.org/abs/1204.0362">The local h-vector of the cluster subdivision of a simplex</a>, arXiv preprint arXiv:1204.0362, 2012
%H A108263 F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999) 73-112.
%H A108263 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 10.
%F A108263 G.f. G=G(t, z) satisfies z*(1+t*z)*G^2 - (1+z)*G + 1 = 0.
%F A108263 T(n, k) = A086810(n-k, k). - _Philippe Deléham_, May 30 2005
%e A108263 T(6,3)=5 because the only short bushes with 6 edges and 3 branchnodes are the five full binary trees with 6 edges.
%e A108263 Triangle begins:
%e A108263 1;
%e A108263 0;
%e A108263 0,1;
%e A108263 0,1;
%e A108263 0,1,2;
%e A108263 0,1,5;
%e A108263 0,1,9,5
%p A108263 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
%p A108263 A108263 := (n,k) -> binomial(n-k-1,n-2*k)*binomial(n,k)/(n-k+1);
%p A108263 seq(print(seq(A108263(n,k),k=0..ceil((n-1)/2))),n=0..8); # _Peter Luschny_, Sep 25 2014
%t A108263 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 *)
%Y A108263 Cf. A005043, A033275, A033276.
%Y A108263 Columns: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279; A005043 (row sums), A033282, A086810
%K A108263 nonn,tabf
%O A108263 0,9
%A A108263 _Emeric Deutsch_, May 29 2005