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A309148
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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.
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11
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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
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1,8
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COMMENTS
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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.
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LINKS
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FORMULA
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A(n,k) = 1/(n*k) * Sum_{d|n} binomial(k*d,d)*(-1)^(n+d)*phi(n/d).
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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A[n_, k_] := 1/(n k) Sum[Binomial[k d, d] (-1)^(n+d) EulerPhi[n/d], {d, Divisors[n]}];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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