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A054619 Triangle T(n,k) = Sum_{d|k} phi(d)*n^(k/d).

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%I A054619 #20 Jan 21 2020 11:58:43
%S A054619 1,2,6,3,12,33,4,20,72,280,5,30,135,660,3145,6,42,228,1344,7800,46956,
%T A054619 7,56,357,2464,16835,118104,823585,8,72,528,4176,32800,262800,2097200,
%U A054619 16781472,9,90,747,6660,59085,532350,4783023,43053480,387422001
%N A054619 Triangle T(n,k) = Sum_{d|k} phi(d)*n^(k/d).
%H A054619 Alois P. Heinz, <a href="/A054619/b054619.txt">Rows n = 1..141, flattened</a>
%e A054619 1;
%e A054619 2, 6;
%e A054619 3, 12, 33;
%e A054619 4, 20, 72,  280;
%e A054619 5, 30, 135, 660,  3145;
%e A054619 6, 42, 228, 1344, 7800, 46956;
%e A054619 ...
%p A054619 with(numtheory):
%p A054619 T:= (n, k)-> add(phi(d)*n^(k/d), d=divisors(k)):
%p A054619 seq(seq(T(n,k), k=1..n), n=1..10);  # _Alois P. Heinz_, Aug 28 2013
%t A054619 T[n_, k_] := Sum[EulerPhi[d]*n^(k/d), {d, Divisors[k]}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 25 2015 *)
%o A054619 (PARI) T(n, k) = sumdiv(k, d, eulerphi(d)*n^(k/d)); \\ _Michel Marcus_, Feb 25 2015
%Y A054619 Cf. A054618, A054630, A054631, A185651 (transpose).
%Y A054619 Main diagonal gives: A228640.
%K A054619 nonn,tabl
%O A054619 1,2
%A A054619 _N. J. A. Sloane_, Apr 16 2000