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A054618 Triangle T(n,k) = Sum_{d|n} phi(d)*k^(n/d).

%I A054618 #25 Jan 23 2022 08:41:08
%S A054618 1,2,6,3,12,33,4,24,96,280,5,40,255,1040,3145,6,84,780,4200,15810,
%T A054618 46956,7,140,2205,16408,78155,279972,823585,8,288,6672,65840,391320,
%U A054618 1681008,5767328,16781472,9,540,19755,262296,1953405,10078164,40354335,134218800,387422001
%N A054618 Triangle T(n,k) = Sum_{d|n} phi(d)*k^(n/d).
%C A054618 Dirichlet convolution of A000010(n) and k^n. - _Richard L. Ollerton_, May 10 2021
%H A054618 Alois P. Heinz, <a href="/A054618/b054618.txt">Rows n = 1..141, flattened</a>
%F A054618 From _Richard L. Ollerton_, May 10 2021: (Start)
%F A054618 T(n,k) = Sum_{i=1..n} k^gcd(n,i).
%F A054618 T(n,k) = Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))/phi(n/gcd(n,i)). (End)
%e A054618 1;
%e A054618 2, 6;
%e A054618 3, 12, 33;
%e A054618 4, 24, 96,  280;
%e A054618 5, 40, 255, 1040, 3145;
%e A054618 6, 84, 780, 4200, 15810, 46956;
%e A054618 ...
%p A054618 with(numtheory):
%p A054618 T:= (n, k)-> add(phi(d)*k^(n/d), d=divisors(n)):
%p A054618 seq(seq(T(n, k), k=1..n), n=1..10); # _Alois P. Heinz_, Aug 28 2013
%p A054618 A054618 := proc(n, k)
%p A054618     add( numtheory[phi](d)*k^(n/d),d=numtheory[divisors](n)) ;
%p A054618 end proc:
%p A054618 seq(seq(A054618(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Jan 23 2022
%t A054618 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 *)
%o A054618 (PARI) T(n, k) = sumdiv(n, d, eulerphi(d)*k^(n/d)); \\ _Michel Marcus_, Feb 25 2015
%Y A054618 Cf. A054619, A054630, A054631.
%Y A054618 Main diagonal gives: A228640.
%Y A054618 Cf. A000010.
%K A054618 nonn,tabl
%O A054618 1,2
%A A054618 _N. J. A. Sloane_, Apr 16 2000