A318557 Number A(n,k) of n-member subsets of [k*n] whose elements sum to a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 5, 10, 1, 0, 1, 1, 6, 30, 38, 1, 0, 1, 1, 9, 55, 165, 126, 1, 0, 1, 1, 10, 91, 460, 1001, 452, 1, 0, 1, 1, 13, 138, 969, 3876, 6198, 1716, 1, 0, 1, 1, 14, 190, 1782, 10630, 33594, 38760, 6470, 1, 0, 1, 1, 17, 253, 2925, 23751, 118755, 296010, 245157, 24310, 1, 0

Offset: 0

Author

Alois P. Heinz, Aug 28 2018

Keywords

• nonn
• tabl

Comments

The sequence of row n satisfies a linear recurrence with constant coefficients of order A018804(n) for n>0.

Examples

			A(3,2) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}.
A(2,3) = 5: {1,2}, {1,5}, {2,4}, {3,6}, {4,5}.
Square array A(n,k) begins:
  1, 1,    1,     1,      1,       1,       1,        1, ...
  0, 1,    1,     1,      1,       1,       1,        1, ...
  0, 1,    2,     5,      6,       9,      10,       13, ...
  0, 1,   10,    30,     55,      91,     138,      190, ...
  0, 1,   38,   165,    460,     969,    1782,     2925, ...
  0, 1,  126,  1001,   3876,   10630,   23751,    46376, ...
  0, 1,  452,  6198,  33594,  118755,  324516,   749398, ...
  0, 1, 1716, 38760, 296010, 1344904, 4496388, 12271518, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..85, flattened

Crossrefs

Columns k=0-10 give: A000007, A000012, A119358, A318591, A318592, A318593, A318594, A318595, A318596, A318597, A318598.

Rows n=0-10 give: A000012, A057427, A042963 (for k>0), A318624, A318625, A318626, A318627, A318628, A318629, A318630, A318631.

Main diagonal gives A318477.

Cf. A018804, A304482.

Programs

Mathematica

nmax = 11; (* Program not suitable to compute a large number of terms. *)
A[n_, k_] := A[n, k] = Count[Subsets[Range[k n], {n}], s_ /; Divisible[Total[s], k]]; A[0, _] = 1;
Table[A[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Oct 04 2019 *)