A070925 Number of subsets of A = {1,2,...,n} that have the same center of gravity as A, i.e., (n+1)/2.

1, 1, 3, 3, 7, 7, 19, 17, 51, 47, 151, 137, 471, 427, 1519, 1391, 5043, 4651, 17111, 15883, 59007, 55123, 206259, 193723, 729095, 688007, 2601639, 2465133, 9358943, 8899699, 33904323, 32342235, 123580883, 118215779, 452902071, 434314137, 1667837679, 1602935103

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 20 2002

Keywords

• nonn

Comments

From Gus Wiseman, Apr 15 2023: (Start)

Also the number of nonempty subsets of {0..n} with mean n/2. The a(0) = 1 through a(5) = 7 subsets are:

{0} {0,1} {1} {0,3} {2} {0,5}

{0,2} {1,2} {0,4} {1,4}

{0,1,2} {0,1,2,3} {1,3} {2,3}

{0,2,4} {0,1,4,5}

{1,2,3} {0,2,3,5}

{0,1,3,4} {1,2,3,4}

{0,1,2,3,4} {0,1,2,3,4,5}

(End)

Examples

			Of the 32 (2^5) sets which can be constructed from the set A = {1,2,3,4,5} only the sets {3}, {2, 3, 4}, {2, 4}, {1, 2, 4, 5}, {1, 2, 3, 4, 5}, {1, 3, 5}, {1, 5} give an average of 3.
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 1..40

Crossrefs

The odd bisection is A000980(n) - 1 = 2*A047653(n) - 1.

For median instead of mean we have A100066, bisection A006134.

Including the empty set gives A222955.

The one-based version is A362046, even bisection A047653(n) - 1.

A007318 counts subsets by length.

A067538 counts partitions with integer mean, strict A102627.

A231147 counts subsets by median.

A327481 counts subsets by integer mean.

Cf. A024718, A057552, A079309, A133406, A326512, A326513, A326537, A327475, A349156, A359893.

Programs

Mathematica

Needs["DiscreteMath`Combinatorica`"]; f[n_] := Block[{s = Subsets[n], c = 0, k = 2}, While[k < 2^n + 1, If[ (Plus @@ s[[k]]) / Length[s[[k]]] == (n + 1)/2, c++ ]; k++ ]; c]; Table[ f[n], {n, 1, 20}]
(* second program *)
Table[Length[Select[Subsets[Range[0,n]],Mean[#]==n/2&]],{n,0,10}] (* Gus Wiseman, Apr 15 2023 *)

Formula

From Gus Wiseman, Apr 18 2023: (Start)

a(2n+1) = A000980(n) - 1.

a(n) = A222955(n) - 1.

a(n) = 2*A362046(n) + 1.

(End)

Extensions

Edited by Robert G. Wilson v and John W. Layman, May 25 2002

a(34)-a(38) from Fausto A. C. Cariboni, Oct 08 2020