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Number of nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= (n-1)/3.
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%I #16 Jun 10 2019 00:24:57

%S 0,0,0,3,6,10,25,45,77,175,322,570,1245,2325,4213,9031,17061,31421,

%T 66547,126763,236203,496063,950818,1787346,3730293,7184421,13598053,

%U 28243063,54604081,103918153,215008363,416990563,797154723

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

%H Robert Israel, <a href="/A048006/b048006.txt">Table of n, a(n) for n = 1..3325</a>

%F a(n) = Sum_{k=1..floor(n/3)} binomial(floor((n-1)/3), 3*k)*binomial(ceiling(2*(n-1)/3), 2*k). - _Robert Israel_, Feb 05 2017

%p f:= n -> add(binomial(floor((n-1)/3),k/3)*binomial(n-floor((n-1)/3),

%p 2*k/3),k=3..n,3):

%p map(f, [$1..100]); # _Robert Israel_, Feb 05 2017

%o (PARI) a(n)=sum(k=1,n\3, binomial((n-1)\3,k)*binomial(n-(n-1)\3,2*k)) \\ _Charles R Greathouse IV_, Feb 05 2017

%K nonn

%O 1,4

%A _Clark Kimberling_