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

1,1

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

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.

Formula

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

Example

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

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 *)

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.

Sequence in context: A309894 A331118 A260723 * A091820 A171922 A306743

Adjacent sequences: A059211 A059212 A059213 * A059215 A059216 A059217

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 15 2001

Status

approved