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%I
%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) = C(n-1, k) + Sum_{i=0..k} C(n, i), k >=1, n >= 0.
%C T(k,n) = maximal number of regions into which k-space can be divided by n hyper-spheres (k >= 1, n >= 0).
%C For all fixed k, the sequences T(k,n) are complete. [_Frank M Jackson_, Jan 26, 2012]
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.
%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.
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