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A341769
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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.
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1
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3, 12, 64, 436, 3624, 35516, 400544, 5106180, 72574936, 1137563980, 19489399824, 362279121044, 7261032943688, 156078126597084, 3581487541784704, 87378336982197028, 2258453972652164280, 61646205047945592428, 1771962416919392083184, 53498826047517147678132
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OFFSET
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1,1
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COMMENTS
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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}.
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LINKS
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FORMULA
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E.g.f.: (1 - x) * e^(2*x) / (2 - e^x)^2.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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