A308667 (1/n) times the number of n-member subsets of [n^2] whose elements sum to a multiple of n.

1, 1, 10, 115, 2126, 54086, 1753074, 69159399, 3220837534, 173103073384, 10551652603526, 719578430425845, 54297978110913252, 4492502634538340722, 404469190271900056316, 39370123445405248353743, 4120204305690280446004838, 461365717080848755611811094

Offset: 1

Author

Alois P. Heinz, Jul 14 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..339
• Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Crossrefs

Main diagonal of A309148.

Cf. A304482, A318557, A318477.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; add(phi(n/d)*
      (-1)^(n+d)*binomial(n*d, d), d=divisors(n))/n^2
    end:
seq(a(n), n=1..20);

Mathematica

a[n_] := a[n] = Sum[EulerPhi[n/d]*
     (-1)^(n + d)*Binomial[n*d, d], {d, Divisors[n]}]/n^2;
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 24 2022, after Alois P. Heinz *)

Formula

a(n) = A309148(n,n).

a(n) = (1/n) * A318477(n).

a(p) == 1 (mod p^3) for all primes p >= 5 (apply Meštrović, Remark 17, p. 12). - Peter Bala, Mar 28 2023

a(n) ~ exp(n - 1/2) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Mar 28 2023