A048094 Number of nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= sqrt(n).

0, 0, 1, 2, 6, 13, 25, 45, 91, 175, 322, 570, 975, 1617, 2607, 7425, 12753, 21385, 35065, 56302, 88638, 136986, 208050, 310840, 1251606, 1973706, 3065931, 4694921, 7092211, 10575635, 15576535, 22674015, 32637675, 46480475

Offset

1,4

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

From Robert Israel, May 25 2026: (Start)

a(n) = Sum_{1 <= k <= s} binomial(s,k) * binomial(n-s,2*k)

= s*(n-s-1)*(n-s)*hypergeom([1, 1-s, -1/2*n+1/2*s+1, -1/2*n+1/2*s+3/2],[3/2,

2, 2],-1)/2

where s = floor(sqrt(n)).

(End)

Maple

f:= proc(n) local s, m;
   s:= floor(sqrt(n));
   add(binomial(s, m) * binomial(n-s, 2*m), m=1..s)
end proc:
map(f, [$1..100]); # Robert Israel, May 25 2026

Crossrefs

Sequence in context: A281865 A267698 A065220 * A335439 A181522 A031872

Adjacent sequences: A048091 A048092 A048093 * A048095 A048096 A048097

Keyword

nonn

Author

Clark Kimberling

Status

approved