

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(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)


LINKS

Table of n, a(n) for n=1..33.


CROSSREFS

Sequence in context: A006017 A147323 A047183 * A048039 A339617 A131913
Adjacent sequences: A047191 A047192 A047193 * A047195 A047196 A047197


KEYWORD

nonn


AUTHOR

Clark Kimberling


STATUS

approved



