A048062 Number of nonempty subsets of {1,2,...,n} in which exactly 2/3 of the elements are <= (n-4)/2.

0, 0, 0, 0, 0, 0, 0, 6, 7, 21, 24, 76, 90, 270, 325, 945, 1155, 3311, 4102, 11611, 14547, 40755, 51555, 143227, 182703, 503943, 647548, 1775092, 2295646, 6259162, 8141055, 22092135, 28881463

Offset

1,8

Links

Robert Israel, Table of n, a(n) for n = 1..3592

Formula

From Robert Israel, Feb 27 2020: (Start)

a(12*k) = 6*(3*k+1)*(3*k-1)*(2*k-1)*hypergeom([1, -6*k-1, 2-3*k, -3*k+5/2],[3/2, 2, 2],-1) for k >= 1.

a(12*k+1) = 9*(3*k-1)*(1+2*k)*(2*k-1)*hypergeom([1, 2-3*k, -6*k-2, -3*k+5/2],[3/2, 2, 2],-1) for k >= 1.

a(12*k+2) = 3*(-1+6*k)*(3*k-1)*(1+2*k)*hypergeom([1, 2-3*k, -6*k-2, -3*k+3/2],[3/2, 2, 2],-1) for k >= 1.

a(12*k+3) = 2*(-1+6*k)*(2+3*k)*(3*k-1)*hypergeom([1, 2-3*k, -6*k-3, -3*k+3/2],[3/2, 2, 2],-1) for k >= 1.

a(12*k+4) = 6*k*(2+3*k)*(-1+6*k)*hypergeom([1, -6*k-3, 1-3*k, -3*k+3/2],[3/2, 2, 2],-1).

a(12*k+5) = 3*k*(5+6*k)*(-1+6*k)*hypergeom([1, 1-3*k, -6*k-4, -3*k+3/2],[3/2, 2, 2],-1).

a(12*k+6) = 3*k*(5+6*k)*(1+6*k)*hypergeom([1, 1-3*k, -6*k-4, -3*k+1/2],[3/2, 2, 2],-1).

a(12*k+7) = 18*k*(k+1)*(1+6*k)*hypergeom([1, 1-3*k, -6*k-5, -3*k+1/2],[3/2, 2, 2],-1).

a(12*k+8) = 6*(3*k+1)*(1+6*k)*(k+1)*hypergeom([1, -3*k, -6*k-5, -3*k+1/2],[3/2, 2, 2],-1).

a(12*k+9) = (3*k+1)*(6*k+7)*(1+6*k)*hypergeom([1, -3*k, -6-6*k, -3*k+1/2],[3/2, 2, 2],-1).

a(12*k+10) = 3*(1+2*k)*(3*k+1)*(6*k+7)*hypergeom([1, -3*k, -6-6*k, -3*k-1/2],[3/2, 2, 2],-1).

a(12*k+11) = 6*(1+2*k)*(4+3*k)*(3*k+1)*hypergeom([1, -3*k, -6*k-7, -3*k-1/2],[3/2, 2, 2],-1). (End)

Maple

G[0]:= i -> 6*(3*i+1)*(3*i-1)*(2*i-1)*hypergeom([1, -6*i-1, 2-3*i, -3*i+5/2], [3/2, 2, 2], -1):
G[1]:= i -> 9*(3*i-1)*(1+2*i)*(2*i-1)*hypergeom([1, 2-3*i, -6*i-2, -3*i+5/2], [3/2, 2, 2], -1):
G[2]:= i -> 3*(-1+6*i)*(3*i-1)*(1+2*i)*hypergeom([1, 2-3*i, -6*i-2, -3*i+3/2], [3/2, 2, 2], -1):
G[3]:= i -> 2*(-1+6*i)*(2+3*i)*(3*i-1)*hypergeom([1, 2-3*i, -6*i-3, -3*i+3/2], [3/2, 2, 2], -1):
G[4]:= i -> 6*i*(2+3*i)*(-1+6*i)*hypergeom([1, 1-3*i, -6*i-3, -3*i+3/2], [3/2, 2, 2], -1):
G[5]:= i -> 3*i*(5+6*i)*(-1+6*i)*hypergeom([1, -6*i-4, 1-3*i, -3*i+3/2], [3/2, 2, 2], -1):
G[6]:= i -> 3*i*(5+6*i)*(1+6*i)*hypergeom([1, -6*i-4, 1-3*i, 1/2-3*i], [3/2, 2, 2], -1):
G[7]:= i -> 18*i*(i+1)*(1+6*i)*hypergeom([1, -5-6*i, 1-3*i, 1/2-3*i], [3/2, 2, 2], -1):
G[8]:= i -> 6*(3*i+1)*(1+6*i)*(i+1)*hypergeom([1, -3*i, -5-6*i, 1/2-3*i], [3/2, 2, 2], -1):
G[9]:= i -> (3*i+1)*(6*i+7)*(1+6*i)*hypergeom([1, -3*i, -6-6*i, 1/2-3*i], [3/2, 2, 2], -1):
G[10]:= i -> 3*(1+2*i)*(3*i+1)*(6*i+7)*hypergeom([1, -3*i, -6-6*i, -1/2-3*i], [3/2, 2, 2], -1):
G[11]:= i -> 6*(1+2*i)*(4+3*i)*(3*i+1)*hypergeom([1, -3*i, -6*i-7, -1/2-3*i], [3/2, 2, 2], -1):
f:= n -> simplify(G[n mod 12](floor(n/12))):
for i from 1 to 3 do f(i):= 0 od:
map(f, [$1..100]); # Robert Israel, Feb 27 2020

Crossrefs

Sequence in context: A042757 A257312 A062369 * A295729 A081284 A185509

Adjacent sequences: A048059 A048060 A048061 * A048063 A048064 A048065

Keyword

nonn

Author

Clark Kimberling

Status

approved