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

1,3

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.

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

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 *)

Crossrefs

Main diagonal of A309148.

Cf. A304482, A318557, A318477.

Sequence in context: A079678 A233908 A089833 * A251318 A083448 A024129

Adjacent sequences: A308664 A308665 A308666 * A308668 A308669 A308670

Keyword

nonn

Author

Alois P. Heinz, Jul 14 2019

Status

approved