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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.

%I A185651 #49 Feb 22 2022 15:17:10
%S A185651 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,
%T A185651 72,96,40,6,0,0,7,42,135,280,255,84,7,0,0,8,56,228,660,1040,780,140,8,
%U A185651 0,0,9,72,357,1344,3145,4200,2205,288,9,0
%N A185651 A(n,k) = Sum_{d|n} phi(d)*k^(n/d); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A185651 Dirichlet convolution of phi(n) and k^n. - _Richard L. Ollerton_, May 07 2021
%H A185651 Alois P. Heinz, <a href="/A185651/b185651.txt">Antidiagonals n = 0..140, flattened</a>
%F A185651 A(n,k) = Sum_{d|n} phi(d)*k^(n/d).
%F A185651 A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258170(n,i). - _Alois P. Heinz_, May 22 2015
%F A185651 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
%F A185651 From _Richard L. Ollerton_, May 07 2021: (Start)
%F A185651 A(n,k) = Sum_{i=1..n} k^gcd(n,i).
%F A185651 A(n,k) = Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))/phi(n/gcd(n,i)).
%F A185651 A(n,k) = A075195(n,k)*n for n >= 1, k >= 1.  (End)
%e A185651 Square array A(n,k) begins:
%e A185651   0, 0,  0,   0,    0,     0,     0, ...
%e A185651   0, 1,  2,   3,    4,     5,     6, ...
%e A185651   0, 2,  6,  12,   20,    30,    42, ...
%e A185651   0, 3, 12,  33,   72,   135,   228, ...
%e A185651   0, 4, 24,  96,  280,   660,  1344, ...
%e A185651   0, 5, 40, 255, 1040,  3145,  7800, ...
%e A185651   0, 6, 84, 780, 4200, 15810, 46956, ...
%p A185651 with(numtheory):
%p A185651 A:= (n, k)-> add(phi(d)*k^(n/d), d=divisors(n)):
%p A185651 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A185651 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 *)
%Y A185651 Columns k=0..10 give A000004, A001477, A053635, A054610, A054611, A054612, A054613, A054614, A054615, A054616, A054617.
%Y A185651 Rows n=0..10 give A000004, A001477, A002378, A054602, A054603, A054604, A054605, A054606, A054607, A054608, A054609.
%Y A185651 Main diagonal gives A228640.
%Y A185651 Cf. A000010, A075195, A258170.
%K A185651 nonn,tabl
%O A185651 0,8
%A A185651 _Alois P. Heinz_, Aug 29 2013