A059214 Square array T(k,n) = C(n-1,k) + Sum_{i=0..k} C(n,i) read by antidiagonals (k >= 1, n >= 1).

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

Offset: 1

Author

N. J. A. Sloane, Feb 15 2001

Keywords

• nonn
• tabl

Comments

For k > 1, gives maximal number of regions into which k-space can be divided by n hyperspheres.

The maximum number of subsets of a set of n points in k-space that can be formed by intersecting it with a hyperplane. - Günter Rote, Dec 18 2018

Examples

			Array begins
   2 4 6  8 10 12 ...
   2 4 8 14 22 32 ...
   2 4 8 16 30 52 ...
		

References

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

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened).
• E. F. Harding, The number of partitions of a set of n points in k dimensions induced by hyperplanes, Proc. Edinburgh Math. Soc., 15 (1967), 285-289.

Crossrefs

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

Equals twice A216274.

Apart from left border, same as A059250. A178522 is probably the best version.

Programs

Mathematica

A059214[k_,n_]:=Binomial[n-1,k]+Sum[Binomial[n,i],{i,0,k}];
Table[A059214[k-n+1,n],{k,10},{n,k}] (* Paolo Xausa, Dec 29 2023 *)

Formula

T(k,n) = C(n-1, k) + Sum_{i=0..k} C(n, i).