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.

1, 1, 2, 1, 6, 3, 1, 22, 15, 4, 1, 90, 93, 28, 5, 1, 394, 645, 244, 45, 6, 1, 1806, 4791, 2380, 505, 66, 7, 1, 8558, 37275, 24868, 6345, 906, 91, 8, 1, 41586, 299865, 272188, 85405, 13926, 1477, 120, 9, 1, 206098, 2474025, 3080596, 1204245, 229326, 26845

Offset

1,3

Comments

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

Links

Table of n, a(n) for n=1..52.

E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett 98 (4) (2006) 162-167.

Andrei Asinowski and Toufik Mansour, Separable d-Permutations and Guillotine Partitions, arXiv 0803.3414 [math.CO], 2008; Annals of Combinatorics 14 (1) pp.17-43 Springer, 2010; Abstract

Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.

Jean Cardinal, Stefan Felsner, Topological drawings of complete bipartite graphs, Journal of Computational Geometry 9.1 (2018), 213-246. Also arXiv:1608.08324 [cs.CG], 2016 (The OEIS is referenced in version v1 but not in v2).

Formula

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.

G.f. of d-th column: [1-z-(z^2-4dz+2z+1)^(1/2)]/(2dz-2z).

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

T(n, k) = hypergeom([-n, n+1], [2], -k). - Peter Luschny, May 23 2014

Example

1,...1,....1,.....1,......1,......1,.......1,.......1,.......1,
1,...2,....3,.....4,......5,......6,.......7,.......8,.......9,
1,...6,...15,....28,.....45,.....66,......91,.....120,.....153,
1,..22,...93,...244,....505,....906,....1477,....2248,....3249,
1,..90,..645,..2380,...6345,..13926,...26845,...47160,...77265,
1,.394,.4791,.24868,..85405,.229326,..522739,.1059976,.1968633,
1,1806,37275,272188,1204245,3956106,10663471,24958200,52546473,

Maple

T := (n, k) -> hypergeom([-n, n+1], [2], -k);
seq(print(seq(simplify(T(n, k)), k=0..9)), n=0..6); # Peter Luschny, May 23 2014

Mathematica

T[0, _] = T[_, 0] = 1;
T[n_, k_] := Sum[Binomial[n+j, 2j] k^j CatalanNumber[j], {j, 0, n}];
Table[T[n-k+1, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018, after Paul Barry *)

Crossrefs

Second column is A006318 (Schroeder numbers), others are A103210 and A103211. Main diagonal is A292798, diagonal under the main diagonal is A103212.

Sequence in context: A163269 A103905 A270967 * A089900 A138533 A173333

Adjacent sequences: A103206 A103207 A103208 * A103210 A103211 A103212

Keyword

nonn,tabl

Author

Ralf Stephan, Jan 27 2005

Status

approved