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

0,3

Links

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

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

Example

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}.

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 *)

Crossrefs

Main diagonal of A304482 and of A318557.

Cf. A119358, A169888, A308667.

Sequence in context: A134362 A296980 A219706 * A219869 A072976 A143414

Adjacent sequences: A318474 A318475 A318476 * A318478 A318479 A318480

Keyword

nonn

Author

Alois P. Heinz, Aug 26 2018

Status

approved