A327027 T(n, k) = (1/n) * Sum_{d|n} phi(d) * A241171(n/d, k) for n >= 1, T(0, k) = 0^k. Triangle read by rows for 0 <= k <= n.

1, 0, 1, 0, 1, 3, 0, 1, 10, 30, 0, 1, 33, 315, 630, 0, 1, 102, 2646, 15120, 22680, 0, 1, 348, 21135, 263340, 1039500, 1247400, 0, 1, 1170, 167310, 4118400, 32432400, 97297200, 97297200, 0, 1, 4113, 1333080, 61757010, 871620750, 4937832900, 11918907000, 10216206000

Offset: 0

Author

Peter Luschny, Aug 20 2019

Keywords

• nonn
• tabl

Comments

We assume A241171 extended to its (0, 0)-based form.

Examples

			[0] 1;
[1] 0, 1;
[2] 0, 1, 3;
[3] 0, 1, 10,   30;
[4] 0, 1, 33,   315,    630;
[5] 0, 1, 102,  2646,   15120,   22680;
[6] 0, 1, 348,  21135,  263340,  1039500,  1247400;
[7] 0, 1, 1170, 167310, 4118400, 32432400, 97297200, 97297200;
		

Crossrefs

Cf. A327021 (main diagonal), A327026 (row sums), A241171, A327029.

Programs

Maple

A327027 := (n, k)-> `if`(n=0, 1, (1/n)*add(phi(d)*A241171(n/d, k), d=divisors(n))):
seq(seq(A327027(n,k), k=0..n), n=0..6);

Mathematica

A327027[0,  k_] := 1;
A327027[n_, k_] := DivisorSum[n, EulerPhi[#] A241171[n/#, k] &] / n;
Table[A327027[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

Sage

# uses[DivisorTriangle from A327029, A241171]
DivisorTriangle(euler_phi, A241171, 8, lambda n: 1/n if n > 1 else 1)