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 A047194 Number of nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= n/3. 1
 0, 0, 1, 3, 6, 13, 25, 45, 91, 175, 322, 645, 1245, 2325, 4651, 9031, 17061, 34123, 66547, 126763, 253527, 496063, 950818, 1901637, 3730293, 7184421, 14368843, 28243063, 54604081, 109208163, 215008363, 416990563, 833981127 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Comment from Jinyuan Wang, Oct 19 2019 (Start) This is also the number of nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= (n+1)/3. Proof: Let b(n) = number of nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= (n+1)/3. We only need to prove b(3k-1) = a(3k-1). Now a(3k-1) = Sum_{m=1..k-1} binomial(k-1, m)*binomial(2k, 2m). b(3k-1) = Sum_{m=1..k-1} binomial(k, m)*binomial(2k-1, 2m). Because binomial(k-1, m)*binomial(2k, 2m) = binomial(k, m)*binomial(2k-1, 2m), we have b(3k-1) = a(3k-1).  (End) LINKS CROSSREFS Sequence in context: A006017 A147323 A047183 * A048039 A339617 A131913 Adjacent sequences:  A047191 A047192 A047193 * A047195 A047196 A047197 KEYWORD nonn AUTHOR STATUS approved

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Last modified May 16 23:46 EDT 2021. Contains 343957 sequences. (Running on oeis4.)