A327028 T(n, k) = k! * Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, k) = 0^k. Triangle read by rows for 0 <= k <= n.
1, 0, 1, 0, 2, 2, 0, 3, 2, 6, 0, 4, 6, 6, 24, 0, 5, 4, 12, 24, 120, 0, 6, 12, 24, 48, 120, 720, 0, 7, 6, 24, 72, 240, 720, 5040, 0, 8, 16, 36, 144, 360, 1440, 5040, 40320, 0, 9, 12, 54, 144, 600, 2160, 10080, 40320, 362880
Offset: 0
Examples
[0] 1 [1] 0, 1 [2] 0, 2, 2 [3] 0, 3, 2, 6 [4] 0, 4, 6, 6, 24 [5] 0, 5, 4, 12, 24, 120 [6] 0, 6, 12, 24, 48, 120, 720 [7] 0, 7, 6, 24, 72, 240, 720, 5040 [8] 0, 8, 16, 36, 144, 360, 1440, 5040, 40320 [9] 0, 9, 12, 54, 144, 600, 2160, 10080, 40320, 362880
Programs
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Maple
A327028 := (n,k) -> `if`(n=0, 1, k!*add(phi(d)*A008284(n/d, k), d = divisors(n))): seq(seq(A327028(n, k), k=0..n), n=0..9);
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Mathematica
A327028[0 , k_] := 1; A327028[n_, k_] := DivisorSum[n, EulerPhi[#] A318144[n/#, k] (-1)^k &]; Table[A327028[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
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SageMath
# uses[DivisorTriangle from A327029] from sage.combinat.partition import number_of_partitions_length def A318144Abs(n, k): return number_of_partitions_length(n, k)*factorial(k) DivisorTriangle(euler_phi, A318144Abs, 10)