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A309148 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.

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