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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.
2
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
0,5
EXAMPLE
[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
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
PROG
(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)
CROSSREFS
Cf. A008284, A318144, A000142 (main diagonal), A327025 (row sums), A327029.
Sequence in context: A129236 A127465 A339033 * A271707 A341445 A360048
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Aug 20 2019
STATUS
approved