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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 29 2013
STATUS
approved