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

1,3

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

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.

Formula

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

Example

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

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 *)

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.

Sequence in context: A327844 A243851 A168266 * A303696 A131074 A059268

Adjacent sequences: A059247 A059248 A059249 * A059251 A059252 A059253

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 15 2001

Extensions

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

Status

approved