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%I #28 Dec 29 2023 13:54:28
%S 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,
%T 8,16,32,52,44,16,2,4,8,16,32,62,84,58,18,2,4,8,16,32,64,114,128,74,
%U 20,2,4,8,16,32,64,126,198,186,92,22,2,4,8,16,32,64
%N Square array T(k,n) = C(n-1,k) + Sum_{i=0..k} C(n,i) read by antidiagonals (k >= 1, n >= 1).
%C For k > 1, gives maximal number of regions into which k-space can be divided by n hyperspheres.
%C 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
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.
%H Paolo Xausa, <a href="/A059214/b059214.txt">Table of n, a(n) for n = 1..11325</a> (antidiagonals 1..150 of the array, flattened).
%H E. F. Harding, <a href="https://doi.org/10.1017/S0013091500011925">The number of partitions of a set of n points in k dimensions induced by hyperplanes</a>, Proc. Edinburgh Math. Soc., 15 (1967), 285-289.
%F T(k,n) = C(n-1, k) + Sum_{i=0..k} C(n, i).
%e Array begins
%e 2 4 6 8 10 12 ...
%e 2 4 8 14 22 32 ...
%e 2 4 8 16 30 52 ...
%t A059214[k_,n_]:=Binomial[n-1,k]+Sum[Binomial[n,i],{i,0,k}];
%t Table[A059214[k-n+1,n],{k,10},{n,k}] (* _Paolo Xausa_, Dec 29 2023 *)
%Y Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).
%Y Equals twice A216274.
%Y Apart from left border, same as A059250. A178522 is probably the best version.
%K nonn,tabl
%O 1,1
%A _N. J. A. Sloane_, Feb 15 2001
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