cp's OEIS Frontend

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.

%I A318557 #21 Oct 04 2019 08:49:48
%S A318557 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,
%T A318557 165,126,1,0,1,1,10,91,460,1001,452,1,0,1,1,13,138,969,3876,6198,1716,
%U A318557 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
%N 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.
%C A318557 The sequence of row n satisfies a linear recurrence with constant coefficients of order A018804(n) for n>0.
%H A318557 Alois P. Heinz, <a href="/A318557/b318557.txt">Antidiagonals n = 0..85, flattened</a>
%e A318557 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}.
%e A318557 A(2,3) = 5: {1,2}, {1,5}, {2,4}, {3,6}, {4,5}.
%e A318557 Square array A(n,k) begins:
%e A318557   1, 1,    1,     1,      1,       1,       1,        1, ...
%e A318557   0, 1,    1,     1,      1,       1,       1,        1, ...
%e A318557   0, 1,    2,     5,      6,       9,      10,       13, ...
%e A318557   0, 1,   10,    30,     55,      91,     138,      190, ...
%e A318557   0, 1,   38,   165,    460,     969,    1782,     2925, ...
%e A318557   0, 1,  126,  1001,   3876,   10630,   23751,    46376, ...
%e A318557   0, 1,  452,  6198,  33594,  118755,  324516,   749398, ...
%e A318557   0, 1, 1716, 38760, 296010, 1344904, 4496388, 12271518, ...
%t A318557 nmax = 11; (* Program not suitable to compute a large number of terms. *)
%t A318557 A[n_, k_] := A[n, k] = Count[Subsets[Range[k n], {n}], s_ /; Divisible[Total[s], k]]; A[0, _] = 1;
%t A318557 Table[A[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Oct 04 2019 *)
%Y A318557 Columns k=0-10 give: A000007, A000012, A119358, A318591, A318592, A318593, A318594, A318595, A318596, A318597, A318598.
%Y A318557 Rows n=0-10 give: A000012, A057427, A042963 (for k>0), A318624, A318625, A318626, A318627, A318628, A318629, A318630, A318631.
%Y A318557 Main diagonal gives A318477.
%Y A318557 Cf. A018804, A304482.
%K A318557 nonn,tabl
%O A318557 0,13
%A A318557 _Alois P. Heinz_, Aug 28 2018