A059250 Square array read by antidiagonals: T(k,n) = binomial(n-1, k) + Sum_{i=0..k} binomial(n, i), k >= 1, n >= 0.

1, 1, 2, 1, 2, 4, 1, 2, 4, 6, 1, 2, 4, 8, 8, 1, 2, 4, 8, 14, 10, 1, 2, 4, 8, 16, 22, 12, 1, 2, 4, 8, 16, 30, 32, 14, 1, 2, 4, 8, 16, 32, 52, 44, 16, 1, 2, 4, 8, 16, 32, 62, 84, 58, 18, 1, 2, 4, 8, 16, 32, 64, 114, 128, 74, 20, 1, 2, 4, 8, 16, 32, 64, 126, 198, 186, 92, 22, 1, 2, 4, 8, 16, 32, 64

Offset: 1

Author

N. J. A. Sloane, Feb 15 2001

Keywords

• nonn
• tabl

Comments

T(k,n) = maximal number of regions into which k-space can be divided by n hyperspheres (k >= 1, n >= 0).

For all fixed k, the sequences T(k,n) are complete. - Frank M Jackson, Jan 26 2012

T(k-1,n) is also the number of regions created by n generic hyperplanes through the origin in k-space (k >= 2). - Kent E. Morrison, Nov 11 2017

Examples

			Array begins
  1, 2, 4, 6,  8, 10, 12, ...
  1, 2, 4, 8, 14, 22, ...
  1, 2, 4, 8, 16, ...
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• K. E. Morrison, From bocce to positivity: some probabilistic linear algebra, arXiv:1405.2994 [math.PR], 2014; Math. Mag., 86 (2013) 110-119.
• L. Schläfli, Theorie der vielfachen Kontinuität, 1901. (See p. 41)
• J. G. Wendel, A problem in geometric probability, Math. Scand., 11 (1962) 109-111.

Crossrefs

Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).

Apart from border, same as A059214. If the k=0 row is included, same as A178522.

Programs

Mathematica

getvalue[n_, k_] := If[n==0, 1, Binomial[n-1, k]+Sum[Binomial[n, i],{i, 0,k}]]; lexicographicLattice[{dim_, maxHeight_}] := Flatten[Array[Sort@Flatten[(Permutations[#1] &) /@     IntegerPartitions[#1 + dim - 1, {dim}], 1] &, maxHeight], 1]; pairs=lexicographicLattice[{2, 13}]-1; Table[getvalue[First[pairs[[j]]], Last[pairs[[j]]]+1], {j, 1, Length[pairs]}] (* Frank M Jackson, Mar 16 2013 *)

Formula

T(k,n) = 2 * Sum_{i=0..k-1} binomial(n-1, i), k >= 1, n >= 1. - Kent E. Morrison, Nov 11 2017

Extensions

Corrected and edited by N. J. A. Sloane, Aug 31 2011, following a suggestion from Frank M Jackson