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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).
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%I #25 Apr 17 2021 08:12:57

%S 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 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 224,629,1050,735,144,9,4,0,1,10,66,352,1300,3162,4102,2224,270,10,10

%N 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 G.f. of column k: Sum_{j>=1} phi(j) * x^j / (1 - k*x^j).

%F T(n,k) = A185651(n,k)/k for k > 0.

%F T(n,k) = Sum_{d|n} phi(n/d)*k^(d - 1).

%e Square array begins:

%e 0, 0, 0, 0, 0, 0, 0, ...

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, 7, ...

%e 2, 3, 6, 11, 18, 27, 38, ...

%e 2, 4, 12, 32, 70, 132, 224, ...

%e 4, 5, 20, 85, 260, 629, 1300, ...

%e 2, 6, 42, 260, 1050, 3162, 7826, ...

%t 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 (PARI) T(n, k) = sum(j=1, n, k^(gcd(j, n)-1));

%o (PARI) T(n, k) = if(n==0, 0, sumdiv(n, d, eulerphi(n/d)*k^(d-1)));

%Y Columns k=0..5 give A000010, A001477, A034738, A034754, A343490, A343492.

%Y Main diagonal gives A056665.

%Y Cf. A185651, A319082.

%K nonn,tabl

%O 0,9

%A _Seiichi Manyama_, Apr 17 2021