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

Author

Ralf Stephan, Jan 27 2005

Keywords

• nonn
• tabl

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

Examples

			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,
		

Links

• 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).

Crossrefs

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

Programs

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 *)

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