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