A309148 A(n,k) is (1/k) times the number of n-member subsets of [k*n] whose elements sum to a multiple of n; square array A(n,k), n>=1, k>=1, read by antidiagonals.

1, 1, 0, 1, 1, 1, 1, 2, 4, 0, 1, 3, 10, 9, 1, 1, 4, 19, 42, 26, 0, 1, 5, 31, 115, 201, 76, 1, 1, 6, 46, 244, 776, 1028, 246, 0, 1, 7, 64, 445, 2126, 5601, 5538, 809, 1, 1, 8, 85, 734, 4751, 19780, 42288, 30666, 2704, 0, 1, 9, 109, 1127, 9276, 54086, 192130, 328755, 173593, 9226, 1

Offset

1,8

Comments

For k > 1 also (1/(k-1)) times the number of n-member subsets of [k*n-1] whose elements sum to a multiple of n.

The sequence of row n satisfies a linear recurrence with constant coefficients of order n.

Links

Alois P. Heinz, Rows n = 1..150, flattened

Formula

A(n,k) = 1/(n*k) * Sum_{d|n} binomial(k*d,d)*(-1)^(n+d)*phi(n/d).

A(n,k) = (1/k) * A304482(n,k).

Example

Square array A(n,k) begins:
  1,   1,    1,     1,      1,      1,       1, ...
  0,   1,    2,     3,      4,      5,       6, ...
  1,   4,   10,    19,     31,     46,      64, ...
  0,   9,   42,   115,    244,    445,     734, ...
  1,  26,  201,   776,   2126,   4751,    9276, ...
  0,  76, 1028,  5601,  19780,  54086,  124872, ...
  1, 246, 5538, 42288, 192130, 642342, 1753074, ...

Maple

with(numtheory):
A:= (n, k)-> add(binomial(k*d, d)*(-1)^(n+d)*
             phi(n/d), d in divisors(n))/(n*k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

Mathematica

A[n_, k_] := 1/(n k) Sum[Binomial[k d, d] (-1)^(n+d) EulerPhi[n/d], {d, Divisors[n]}];
Table[A[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 04 2019 *)

Crossrefs

Columns k=1-10 give: A000035, A145855, A309182, A309183, A309184, A309185, A309186, A309187, A309188, A309189.

Rows n=1-3 give: A000012, A001477(k-1), A005448.

Main diagonal gives A308667.

Cf. A000010, A304482.

Sequence in context: A378323 A378290 A118343 * A351761 A226031 A308460

Adjacent sequences: A309145 A309146 A309147 * A309149 A309150 A309151

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 14 2019

Status

approved