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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. 11
 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, 6, 46, 244, 776, 1028, 246, 0, 1, 7, 64, 445, 2126, 5601, 5538, 809, 1, 1, 8, 85, 734, 4751, 19780, 42288, 30666, 2704, 0, 1, 9, 109, 1127, 9276, 54086, 192130, 328755, 173593, 9226, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS 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. The sequence of row n satisfies a linear recurrence with constant coefficients of order n. LINKS Alois P. Heinz, Rows n = 1..150, flattened FORMULA A(n,k) = 1/(n*k) * Sum_{d|n} binomial(k*d,d)*(-1)^(n+d)*phi(n/d). A(n,k) = (1/k) * A304482(n,k). EXAMPLE Square array A(n,k) begins:   1,   1,    1,     1,      1,      1,       1, ...   0,   1,    2,     3,      4,      5,       6, ...   1,   4,   10,    19,     31,     46,      64, ...   0,   9,   42,   115,    244,    445,     734, ...   1,  26,  201,   776,   2126,   4751,    9276, ...   0,  76, 1028,  5601,  19780,  54086,  124872, ...   1, 246, 5538, 42288, 192130, 642342, 1753074, ... MAPLE with(numtheory): A:= (n, k)-> add(binomial(k*d, d)*(-1)^(n+d)*              phi(n/d), d in divisors(n))/(n*k): seq(seq(A(n, 1+d-n), n=1..d), d=1..12); MATHEMATICA A[n_, k_] := 1/(n k) Sum[Binomial[k d, d] (-1)^(n+d) EulerPhi[n/d], {d, Divisors[n]}]; Table[A[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 04 2019 *) CROSSREFS Columns k=1-10 give: A000035, A145855, A309182, A309183, A309184, A309185, A309186, A309187, A309188, A309189. Rows n=1-3 give: A000012, A001477(k-1), A005448. Main diagonal gives A308667. Cf. A000010, A304482. Sequence in context: A256245 A173004 A118343 * A226031 A308460 A244116 Adjacent sequences:  A309145 A309146 A309147 * A309149 A309150 A309151 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 14 2019 STATUS approved

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Last modified January 15 19:15 EST 2021. Contains 340189 sequences. (Running on oeis4.)