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A343489 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=1..n} k^(gcd(j, n) - 1).

%I A343489 #25 Apr 17 2021 08:12:57
%S A343489 0,0,1,0,1,1,0,1,2,2,0,1,3,3,2,0,1,4,6,4,4,0,1,5,11,12,5,2,0,1,6,18,
%T A343489 32,20,6,6,0,1,7,27,70,85,42,7,4,0,1,8,38,132,260,260,70,8,6,0,1,9,51,
%U A343489 224,629,1050,735,144,9,4,0,1,10,66,352,1300,3162,4102,2224,270,10,10
%N A343489 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=1..n} k^(gcd(j, n) - 1).
%F A343489 G.f. of column k: Sum_{j>=1} phi(j) * x^j / (1 - k*x^j).
%F A343489 T(n,k) = A185651(n,k)/k for k > 0.
%F A343489 T(n,k) = Sum_{d|n} phi(n/d)*k^(d - 1).
%e A343489 Square array begins:
%e A343489   0, 0,  0,   0,    0,    0,    0, ...
%e A343489   1, 1,  1,   1,    1,    1,    1, ...
%e A343489   1, 2,  3,   4,    5,    6,    7, ...
%e A343489   2, 3,  6,  11,   18,   27,   38, ...
%e A343489   2, 4, 12,  32,   70,  132,  224, ...
%e A343489   4, 5, 20,  85,  260,  629, 1300, ...
%e A343489   2, 6, 42, 260, 1050, 3162, 7826, ...
%t A343489 T[n_, k_] := Sum[If[k == (g = GCD[j, n] - 1) == 0, 1, k^g], {j, 1, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Apr 17 2021 *)
%o A343489 (PARI) T(n, k) = sum(j=1, n, k^(gcd(j, n)-1));
%o A343489 (PARI) T(n, k) = if(n==0, 0, sumdiv(n, d, eulerphi(n/d)*k^(d-1)));
%Y A343489 Columns k=0..5 give A000010, A001477, A034738, A034754, A343490, A343492.
%Y A343489 Main diagonal gives A056665.
%Y A343489 Cf. A185651, A319082.
%K A343489 nonn,tabl
%O A343489 0,9
%A A343489 _Seiichi Manyama_, Apr 17 2021