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%I #29 Oct 04 2019 13:10:05
%S 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,
%T 6,46,244,776,1028,246,0,1,7,64,445,2126,5601,5538,809,1,1,8,85,734,
%U 4751,19780,42288,30666,2704,0,1,9,109,1127,9276,54086,192130,328755,173593,9226,1
%N 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.
%C 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.
%C The sequence of row n satisfies a linear recurrence with constant coefficients of order n.
%H Alois P. Heinz, <a href="/A309148/b309148.txt">Rows n = 1..150, flattened</a>
%F A(n,k) = 1/(n*k) * Sum_{d|n} binomial(k*d,d)*(-1)^(n+d)*phi(n/d).
%F A(n,k) = (1/k) * A304482(n,k).
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, ...
%e 1, 4, 10, 19, 31, 46, 64, ...
%e 0, 9, 42, 115, 244, 445, 734, ...
%e 1, 26, 201, 776, 2126, 4751, 9276, ...
%e 0, 76, 1028, 5601, 19780, 54086, 124872, ...
%e 1, 246, 5538, 42288, 192130, 642342, 1753074, ...
%p with(numtheory):
%p A:= (n, k)-> add(binomial(k*d, d)*(-1)^(n+d)*
%p phi(n/d), d in divisors(n))/(n*k):
%p seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t A[n_, k_] := 1/(n k) Sum[Binomial[k d, d] (-1)^(n+d) EulerPhi[n/d], {d, Divisors[n]}];
%t Table[A[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Oct 04 2019 *)
%Y Columns k=1-10 give: A000035, A145855, A309182, A309183, A309184, A309185, A309186, A309187, A309188, A309189.
%Y Rows n=1-3 give: A000012, A001477(k-1), A005448.
%Y Main diagonal gives A308667.
%Y Cf. A000010, A304482.
%K nonn,tabl
%O 1,8
%A _Alois P. Heinz_, Jul 14 2019
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