A047194 - OEIS

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A047194

Number of nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= n/3.

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

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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)

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