A059250 - OEIS

%I #54 Dec 28 2023 14:49:55

%S 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,

%T 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,

%U 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

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

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

%C For all fixed k, the sequences T(k,n) are complete. - _Frank M Jackson_, Jan 26 2012

%C 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

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

%H G. C. Greubel, <a href="/A059250/b059250.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H K. E. Morrison, <a href="https://arxiv.org/abs/1405.2994">From bocce to positivity: some probabilistic linear algebra</a>, arXiv:1405.2994 [math.PR], 2014; Math. Mag., 86 (2013) 110-119.

%H L. Schläfli, <a href="https://books.google.com/books?id=foIUAQAAMAAJ"> Theorie der vielfachen Kontinuität</a>, 1901. (See p. 41)

%H J. G. Wendel, <a href="http://dx.doi.org/10.7146/math.scand.a-10655">A problem in geometric probability</a>, Math. Scand., 11 (1962) 109-111.

%F T(k,n) =  2 * Sum_{i=0..k-1} binomial(n-1, i), k >= 1, n >= 1. - _Kent E. Morrison_, Nov 11 2017

%e Array begins

%e   1, 2, 4, 6,  8, 10, 12, ...

%e   1, 2, 4, 8, 14, 22, ...

%e   1, 2, 4, 8, 16, ...

%t 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 *)

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

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

%K nonn,tabl

%O 1,3

%A _N. J. A. Sloane_, Feb 15 2001

%E Corrected and edited by _N. J. A. Sloane_, Aug 31 2011, following a suggestion from _Frank M Jackson_