A309189 -id:A309189

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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.

+10
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

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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.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 14 2019

STATUS

approved

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