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

0,13

Comments

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

Links

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

Example

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, ...

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 *)

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.

Sequence in context: A198062 A347617 A226690 * A245683 A273712 A129634

Adjacent sequences: A318554 A318555 A318556 * A318558 A318559 A318560

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 28 2018

Status

approved