A309128 (1/n) times the sum of the elements of all subsets of [n] whose sum is divisible by n.

1, 1, 4, 4, 12, 20, 40, 70, 150, 284, 564, 1116, 2212, 4392, 8768, 17404, 34704, 69214, 137980, 275264, 549340, 1096244, 2188344, 4369196, 8724196, 17422500, 34797476, 69505628, 138845940, 277383904, 554189344, 1107296248, 2212559996, 4421289872, 8835361488

Offset: 1

Author

Alois P. Heinz, Jul 13 2019

Keywords

• nonn

Examples

			The subsets of [5] whose sum is divisible by 5 are: {}, {5}, {1,4}, {2,3}, {1,4,5}, {2,3,5}, {1,2,3,4}, {1,2,3,4,5}.  The sum of their elements is 0 + 5 + 5 + 5 + 10 + 10 + 10 + 15 = 60.  So a(5) = 60/5 = 12.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000010, A001792 (the same for all subsets), A053636, A063776, A309122, A309280.

Programs

Maple

b:= proc(n, m, s) option remember; `if`(n=0, [`if`(s=0, 1, 0), 0],
      b(n-1, m, s) +(g-> g+[0, g[1]*n])(b(n-1, m, irem(s+n, m))))
    end:
a:= proc(n) option remember; forget(b); b(n$2, 0)[2]/n end:
seq(a(n), n=1..40);

Mathematica

b[n_, m_, s_] := b[n, m, s] = If[n == 0, {If[s == 0, 1, 0], 0},
     b[n-1, m, s] + Function[g, g+{0, g[[1]] n}][b[n-1, m, Mod[s+n, m]]]];
a[n_] := b[n, n, 0][[2]]/n;
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

Formula

Conjecture: a(n) = (n + 1) * A063776(n)/4 - (phi(n)/2) * (1 + (-1)^n)/2 = ((n + 1)/(4*n)) * A053636(n) - (phi(n)/2) * (1 + (-1)^n)/2. - Petros Hadjicostas, Jul 20 2019

a(n) = A309280(n,n). - Alois P. Heinz, Jul 21 2019