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

0,8

Comments

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

Links

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

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)

Example

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, ...

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 *)

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.

Sequence in context: A343042 A343046 A271917 * A265080 A228275 A228250

Adjacent sequences: A185648 A185649 A185650 * A185652 A185653 A185654

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 29 2013

Status

approved