A048095 Number of nonempty subsets of {1,2,...,n} in which exactly 2/3 of the elements are <= sqrt(n).

0, 0, 0, 2, 3, 4, 5, 6, 18, 21, 24, 27, 30, 33, 36, 138, 156, 175, 195, 216, 238, 261, 285, 310, 1150, 1260, 1375, 1495, 1620, 1750, 1885, 2025, 2170, 2320, 2475, 11035, 11935, 12880, 13871, 14909, 15995, 17130, 18315, 19551, 20839, 22180, 23575, 25025, 111377, 118895, 126742, 134925, 143451

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/2} binomial(s,2*k) * binomial(n-s,2*k)

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

Crossrefs

Sequence in context: A337865 A297181 A294121 * A264975 A031015 A329581

Adjacent sequences: A048092 A048093 A048094 * A048096 A048097 A048098

Keyword

nonn,changed

Author

Clark Kimberling

Status

approved