A308667 - OEIS

%I #47 Mar 30 2023 05:32:17

%S 1,1,10,115,2126,54086,1753074,69159399,3220837534,173103073384,

%T 10551652603526,719578430425845,54297978110913252,4492502634538340722,

%U 404469190271900056316,39370123445405248353743,4120204305690280446004838,461365717080848755611811094

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

%H Alois P. Heinz, <a href="/A308667/b308667.txt">Table of n, a(n) for n = 1..339</a>

%H Romeo Meštrović, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.

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

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

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

%F a(n) ~ exp(n - 1/2) * n^(n - 5/2) / sqrt(2*Pi). - _Vaclav Kotesovec_, Mar 28 2023

%p with(numtheory):

%p a:= proc(n) option remember; add(phi(n/d)*

%p       (-1)^(n+d)*binomial(n*d, d), d=divisors(n))/n^2

%p     end:

%p seq(a(n), n=1..20);

%t a[n_] := a[n] = Sum[EulerPhi[n/d]*

%t      (-1)^(n + d)*Binomial[n*d, d], {d, Divisors[n]}]/n^2;

%t Table[a[n], {n, 1, 20}] (* _Jean-François Alcover_, Mar 24 2022, after _Alois P. Heinz_ *)

%Y Main diagonal of A309148.

%Y Cf. A304482, A318557, A318477.

%K nonn

%O 1,3

%A _Alois P. Heinz_, Jul 14 2019