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A214283
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Smallest Euler characteristic of a downset on an n-dimensional cube.
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8
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0, -1, -2, -3, -4, -10, -20, -35, -56, -126, -252, -462, -792, -1716, -3432, -6435, -11440, -24310, -48620, -92378, -167960, -352716, -705432, -1352078, -2496144, -5200300, -10400600, -20058300, -37442160, -77558760, -155117520, -300540195
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OFFSET
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1,3
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COMMENTS
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An m-downset is a set of subsets of 1..m such that if S is in the set, so are all subsets of S. The Euler characteristic of a downset is the number of sets in the downset with an even cardinality, minus the number with an odd cardinality.
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LINKS
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FORMULA
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a(n=2k) = -binomial(n-1,n/2) = -binomial(2k-1,k),
a(n=4k+3) = -binomial(n-1,(n-1)/2) = -binomial(4k+2,2k+1),
a(n=4k+1) = -binomial(n-1,(n+1)/2) = -binomial(4k,2k+1).
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MATHEMATICA
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PROG
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(Haskell)
a214283 1 = 0
a214283 n = - a007318 (n - 1) (a004525 n)
(Python)
from math import comb
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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