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: A384619 A381602 A394901 * A396413 A396447 A351761

Adjacent sequences: A309145 A309146 A309147 * A309149 A309150 A309151

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 14 2019

Status

approved