

A059250


Square array read by antidiagonals: T(k,n) = binomial(n1, k) + Sum_{i=0..k} binomial(n, i), k >= 1, n >= 0.


6



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, 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, 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
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OFFSET

1,3


COMMENTS

T(k,n) = maximal number of regions into which kspace can be divided by n hyperspheres (k >= 1, n >= 0).
For all fixed k, the sequences T(k,n) are complete.  Frank M Jackson, Jan 26 2012
T(k1,n) is also the number of regions created by n generic hyperplanes through the origin in kspace (k >= 2).  Kent E. Morrison, Nov 11 2017


REFERENCES

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


LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
K. E. Morrison, From bocce to positivity: some probabilistic linear algebra, Math. Mag., 86 (2013) 110119.
L. SchlĂ¤fli, Theorie der vielfachen KontinuitĂ¤t, 1901. (See p. 41)
J. G. Wendel, A problem in geometric probability, Math. Scand., 11 (1962) 109111.


FORMULA

T(k,n) = 2 * Sum_{i=0..k1} binomial(n1, i), k >= 1, n >= 1.  Kent E. Morrison, Nov 11 2017


EXAMPLE

Array begins
1, 2, 4, 6, 8, 10, 12, ...
1, 2, 4, 8, 14, 22, ...
1, 2, 4, 8, 16, ...


MATHEMATICA

getvalue[n_, k_] := If[n==0, 1, Binomial[n1, 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 *)


CROSSREFS

Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).
Apart from border, same as A059214. If the k=0 row is included, same as A178522.
Sequence in context: A141539 A243851 A168266 * A303696 A131074 A059268
Adjacent sequences: A059247 A059248 A059249 * A059251 A059252 A059253


KEYWORD

nonn,tabl


AUTHOR

N. J. A. Sloane, Feb 15 2001


EXTENSIONS

Corrected and edited by N. J. A. Sloane, Aug 31 2011, following a suggestion from Frank M Jackson


STATUS

approved



