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

Author

Peter Luschny, Aug 20 2019

Keywords

• nonn
• tabl

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
		

Crossrefs

Cf. A008284, A318144, A000142 (main diagonal), A327025 (row sums), A327029.

Programs

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);

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

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)