A185651 A(n,k) = Sum_{d|n} phi(d)*k^(n/d); square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 20, 33, 24, 5, 0, 0, 6, 30, 72, 96, 40, 6, 0, 0, 7, 42, 135, 280, 255, 84, 7, 0, 0, 8, 56, 228, 660, 1040, 780, 140, 8, 0, 0, 9, 72, 357, 1344, 3145, 4200, 2205, 288, 9, 0

Offset: 0

Author

Alois P. Heinz, Aug 29 2013

Keywords

• nonn
• tabl

Comments

Dirichlet convolution of phi(n) and k^n. - Richard L. Ollerton, May 07 2021

Examples

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,     0, ...
  0, 1,  2,   3,    4,     5,     6, ...
  0, 2,  6,  12,   20,    30,    42, ...
  0, 3, 12,  33,   72,   135,   228, ...
  0, 4, 24,  96,  280,   660,  1344, ...
  0, 5, 40, 255, 1040,  3145,  7800, ...
  0, 6, 84, 780, 4200, 15810, 46956, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Columns k=0..10 give A000004, A001477, A053635, A054610, A054611, A054612, A054613, A054614, A054615, A054616, A054617.

Rows n=0..10 give A000004, A001477, A002378, A054602, A054603, A054604, A054605, A054606, A054607, A054608, A054609.

Main diagonal gives A228640.

Cf. A000010, A075195, A258170.

Programs

Maple

with(numtheory):
A:= (n, k)-> add(phi(d)*k^(n/d), d=divisors(n)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

a[, 0] = a[0, ] = 0; a[n_, k_] := Sum[EulerPhi[d]*k^(n/d), {d, Divisors[n]}]; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)

Formula

A(n,k) = Sum_{d|n} phi(d)*k^(n/d).

A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258170(n,i). - Alois P. Heinz, May 22 2015

G.f. for column k: Sum_{n>=1} phi(n)*k*x^n/(1-k*x^n) for k >= 0. - Petros Hadjicostas, Nov 06 2017

From Richard L. Ollerton, May 07 2021: (Start)

A(n,k) = Sum_{i=1..n} k^gcd(n,i).

A(n,k) = Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))/phi(n/gcd(n,i)).

A(n,k) = A075195(n,k)*n for n >= 1, k >= 1. (End)