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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.

%I A327028 #19 Mar 24 2020 09:00:55
%S A327028 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,
%T A327028 0,7,6,24,72,240,720,5040,0,8,16,36,144,360,1440,5040,40320,0,9,12,54,
%U A327028 144,600,2160,10080,40320,362880
%N 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.
%e A327028 [0] 1
%e A327028 [1] 0, 1
%e A327028 [2] 0, 2,  2
%e A327028 [3] 0, 3,  2,  6
%e A327028 [4] 0, 4,  6,  6,  24
%e A327028 [5] 0, 5,  4, 12,  24, 120
%e A327028 [6] 0, 6, 12, 24,  48, 120,  720
%e A327028 [7] 0, 7,  6, 24,  72, 240,  720,  5040
%e A327028 [8] 0, 8, 16, 36, 144, 360, 1440,  5040, 40320
%e A327028 [9] 0, 9, 12, 54, 144, 600, 2160, 10080, 40320, 362880
%p A327028 A327028 := (n,k) -> `if`(n=0, 1, k!*add(phi(d)*A008284(n/d, k), d = divisors(n))):
%p A327028 seq(seq(A327028(n, k), k=0..n), n=0..9);
%t A327028 A327028[0 , k_] := 1;
%t A327028 A327028[n_, k_] := DivisorSum[n, EulerPhi[#] A318144[n/#, k] (-1)^k &];
%t A327028 Table[A327028[n, k], {n, 0,  9}, {k, 0,  n}] // Flatten
%o A327028 (SageMath) # uses[DivisorTriangle from A327029]
%o A327028 from sage.combinat.partition import number_of_partitions_length
%o A327028 def A318144Abs(n, k): return number_of_partitions_length(n, k)*factorial(k)
%o A327028 DivisorTriangle(euler_phi, A318144Abs, 10)
%Y A327028 Cf. A008284, A318144, A000142 (main diagonal), A327025 (row sums), A327029.
%K A327028 nonn,tabl
%O A327028 0,5
%A A327028 _Peter Luschny_, Aug 20 2019