A318477 Number of n-member subsets of [n^2] whose elements sum to a multiple of n.

1, 1, 2, 30, 460, 10630, 324516, 12271518, 553275192, 28987537806, 1731030733840, 116068178638786, 8634941165110140, 705873715441872276, 62895036883536770108, 6067037854078500844740, 629921975126483973659888, 70043473196734767582082246

Offset: 0

Author

Alois P. Heinz, Aug 26 2018

Keywords

• nonn

Examples

			a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 2: {1,3}, {2,4}.
a(3) = 30: {1,2,3}, {1,2,6}, {1,2,9}, {1,3,5}, {1,3,8}, {1,4,7}, {1,5,6}, {1,5,9}, {1,6,8}, {1,8,9}, {2,3,4}, {2,3,7}, {2,4,6}, {2,4,9}, {2,5,8}, {2,6,7}, {2,7,9}, {3,4,5}, {3,4,8}, {3,5,7}, {3,6,9}, {3,7,8}, {4,5,6}, {4,5,9}, {4,6,8}, {4,8,9}, {5,6,7}, {5,7,9}, {6,7,8}, {7,8,9}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..338

Crossrefs

Main diagonal of A304482 and of A318557.

Cf. A119358, A169888, A308667.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(phi(n/d)*
      (-1)^(n+d)*binomial(n*d, d), d=divisors(n))/n)
    end:
seq(a(n), n=0..20);

Mathematica

a[n_] := (-1)^n Sum[(-1)^d Binomial[d n, d] EulerPhi[n/d], {d, Divisors[n]} ]/n; a[0] = 1;
a /@ Range[0, 20] (* Jean-François Alcover, Sep 23 2019 *)

Formula

a(n) = n * A308667(n) for n >= 1.

a(n) ~ exp(n - 1/2) * n^(n - 3/2) / sqrt(2*Pi). - Vaclav Kotesovec, Mar 28 2023