cp's OEIS Frontend

A103209 Square array T(n,d) read by antidiagonals: number of structurally-different guillotine partitions of a d-dimensional box in R^d by n hyperplanes.

%I A103209 #37 Oct 24 2018 19:13:57
%S A103209 1,1,2,1,6,3,1,22,15,4,1,90,93,28,5,1,394,645,244,45,6,1,1806,4791,
%T A103209 2380,505,66,7,1,8558,37275,24868,6345,906,91,8,1,41586,299865,272188,
%U A103209 85405,13926,1477,120,9,1,206098,2474025,3080596,1204245,229326,26845
%N A103209 Square array T(n,d) read by antidiagonals: number of structurally-different guillotine partitions of a d-dimensional box in R^d by n hyperplanes.
%C A103209 The columns are the row sums of the inverses of the Riordan arrays ((1-d*x)/(1-x),x(1-d*x)/(1-x)), that is, of the Riordan arrays ((1+x-sqrt(1+2(1-2*d)x+x^2)/(2*d*x),(1+x-sqrt(1+2(1-2*d)x+x^2)/(2*d)). - _Paul Barry_, May 24 2005
%H A103209 E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, <a href="http://dx.doi.org/10.1016/j.ipl.2006.01.011">The number of guillotine partitions in d dimensions</a>, Inf. Proc. Lett 98 (4) (2006) 162-167.
%H A103209 Andrei Asinowski and Toufik Mansour, <a href="http://arxiv.org/abs/0803.3414">Separable d-Permutations and Guillotine Partitions</a>, arXiv 0803.3414 [math.CO], 2008; Annals of Combinatorics 14 (1) pp.17-43 Springer, 2010; <a href="http://www.combinatorics.net/new/Annals/Abstract/14_1_17.aspx">Abstract</a>
%H A103209 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry3/barry252.html">On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.6.
%H A103209 Jean Cardinal, Stefan Felsner, <a href="https://doi.org/10.20382/jocg.v9i1a7">Topological drawings of complete bipartite graphs</a>, Journal of Computational Geometry 9.1 (2018), 213-246. Also <a href="https://arxiv.org/abs/1608.08324">arXiv:1608.08324</a> [cs.CG], 2016 (The OEIS is referenced in version v1 but not in v2).
%F A103209 T(n, d) = (1/n) * sum[i=0..n-1, C(n, i)*C(n, i+1)*(d-1)^i*d^(n-i) ], T(n, 0)=1.
%F A103209 G.f. of d-th column: [1-z-(z^2-4dz+2z+1)^(1/2)]/(2dz-2z).
%F A103209 T(n, k) = sum{j=0..n, C(n+j, 2j)*k^j*C(j)}, C(n) as in A000108. - _Paul Barry_, May 21 2005
%F A103209 T(n, k) = hypergeom([-n, n+1], [2], -k). - _Peter Luschny_, May 23 2014
%e A103209 1,...1,....1,.....1,......1,......1,.......1,.......1,.......1,
%e A103209 1,...2,....3,.....4,......5,......6,.......7,.......8,.......9,
%e A103209 1,...6,...15,....28,.....45,.....66,......91,.....120,.....153,
%e A103209 1,..22,...93,...244,....505,....906,....1477,....2248,....3249,
%e A103209 1,..90,..645,..2380,...6345,..13926,...26845,...47160,...77265,
%e A103209 1,.394,.4791,.24868,..85405,.229326,..522739,.1059976,.1968633,
%e A103209 1,1806,37275,272188,1204245,3956106,10663471,24958200,52546473,
%p A103209 T := (n,k) -> hypergeom([-n, n+1], [2], -k);
%p A103209 seq(print(seq(simplify(T(n, k)), k=0..9)), n=0..6); # _Peter Luschny_, May 23 2014
%t A103209 T[0, _] = T[_, 0] = 1;
%t A103209 T[n_, k_] := Sum[Binomial[n+j, 2j] k^j CatalanNumber[j], {j, 0, n}];
%t A103209 Table[T[n-k+1, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 20 2018, after _Paul Barry_ *)
%Y A103209 Second column is A006318 (Schroeder numbers), others are A103210 and A103211. Main diagonal is A292798, diagonal under the main diagonal is A103212.
%K A103209 nonn,tabl
%O A103209 1,3
%A A103209 _Ralf Stephan_, Jan 27 2005