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A341769 Number of connected components of Euclidean n-space when the hyperplanes x_i+x_j=1, x_i=0, x_i=1 (1 <= i < j <= n) are deleted. 1
3, 12, 64, 436, 3624, 35516, 400544, 5106180, 72574936, 1137563980, 19489399824, 362279121044, 7261032943688, 156078126597084, 3581487541784704, 87378336982197028, 2258453972652164280, 61646205047945592428, 1771962416919392083184, 53498826047517147678132 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(n) is also the number of labeled colored threshold graphs on n vertices. Threshold graphs are constructed recursively such that each vertex added is either adjacent to all previous vertices (called dominating vertex) or an isolated vertex. Threshold graphs where either no vertex is colored, or from some vertex onwards in the construction, all dominating vertices are colored red and all isolated vertices are colored blue are called colored threshold graphs. Labeled colored threshold graphs are those with n vertices labeled distinctly using {1,...,n}.
LINKS
P. Deshpande, K. Menon, and A. Singh, Counting regions of the boxed threshold arrangement, arXiv:2101.12060 [math.CO], 2021.
FORMULA
a(n) = 4*A000670(n) + Sum_{k=1..n} 4*(k!-(k-1)!)*(k*A008277(n,k) - n*A008277(n-1,k-1)) for n >= 2.
E.g.f.: (1 - x) * e^(2*x) / (2 - e^x)^2.
MATHEMATICA
MapAt[# - 1 &, Array[2 (PolyLog[-#, 1/2] + KroneckerDelta[#]) + Sum[4 (k! - (k - 1)!) (k StirlingS2[#, k] - # StirlingS2[# - 1, k - 1]), {k, #}] &, 20], 1] (* Michael De Vlieger, May 07 2021 *)
CROSSREFS
Sequence in context: A207557 A235129 A222033 * A302195 A359660 A196559
KEYWORD
nonn
AUTHOR
Krishna Menon, Feb 19 2021
STATUS
approved

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)