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A054618
Triangle T(n,k) = Sum_{d|n} phi(d)*k^(n/d).
6
1, 2, 6, 3, 12, 33, 4, 24, 96, 280, 5, 40, 255, 1040, 3145, 6, 84, 780, 4200, 15810, 46956, 7, 140, 2205, 16408, 78155, 279972, 823585, 8, 288, 6672, 65840, 391320, 1681008, 5767328, 16781472, 9, 540, 19755, 262296, 1953405, 10078164, 40354335, 134218800, 387422001
OFFSET
1,2
COMMENTS
Dirichlet convolution of A000010(n) and k^n. - Richard L. Ollerton, May 10 2021
LINKS
FORMULA
From Richard L. Ollerton, May 10 2021: (Start)
T(n,k) = Sum_{i=1..n} k^gcd(n,i).
T(n,k) = Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))/phi(n/gcd(n,i)). (End)
EXAMPLE
1;
2, 6;
3, 12, 33;
4, 24, 96, 280;
5, 40, 255, 1040, 3145;
6, 84, 780, 4200, 15810, 46956;
...
MAPLE
with(numtheory):
T:= (n, k)-> add(phi(d)*k^(n/d), d=divisors(n)):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Aug 28 2013
A054618 := proc(n, k)
add( numtheory[phi](d)*k^(n/d), d=numtheory[divisors](n)) ;
end proc:
seq(seq(A054618(n, k), k=1..n), n=1..10) ; # R. J. Mathar, Jan 23 2022
MATHEMATICA
T[n_, k_] := Sum[EulerPhi[d]*k^(n/d), {d, Divisors[n]}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 25 2015 *)
PROG
(PARI) T(n, k) = sumdiv(n, d, eulerphi(d)*k^(n/d)); \\ Michel Marcus, Feb 25 2015
CROSSREFS
Main diagonal gives: A228640.
Cf. A000010.
Sequence in context: A352793 A207901 A054619 * A120859 A253258 A098810
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Apr 16 2000
STATUS
approved