

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
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OFFSET

1,4


COMMENTS

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(3k1) = a(3k1).
Now a(3k1) = Sum_{m=1..k1} binomial(k1, m)*binomial(2k, 2m).
b(3k1) = Sum_{m=1..k1} binomial(k, m)*binomial(2k1, 2m).
Because binomial(k1, m)*binomial(2k, 2m) = binomial(k, m)*binomial(2k1, 2m), we have b(3k1) = a(3k1). (End)


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KEYWORD

nonn


AUTHOR



STATUS

approved



