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A185651 A(n,k) = Sum_{d|n} phi(d)*k^(n/d); square array A(n,k), n>=0, k>=0, read by antidiagonals. 26
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
Dirichlet convolution of phi(n) and k^n. - Richard L. Ollerton, May 07 2021
LINKS
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
Main diagonal gives A228640.
Sequence in context: A343042 A343046 A271917 * A265080 A228275 A228250
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 29 2013
STATUS
approved

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Last modified May 18 12:17 EDT 2024. Contains 372630 sequences. (Running on oeis4.)