%I #26 May 11 2021 01:55:03

%S 1,1,2,1,3,2,1,5,4,3,1,9,10,9,2,1,17,28,33,6,4,1,33,82,129,26,24,2,1,

%T 65,244,513,126,182,8,4,1,129,730,2049,626,1458,50,41,3,1,257,2188,

%U 8193,3126,11954,344,577,37,4,1,513,6562,32769,15626,99594,2402,8705,811,68,2

%N Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n/d).

%H Seiichi Manyama, <a href="/A308509/b308509.txt">Antidiagonals n = 1..140, flattened</a>

%F L.g.f. of column k: -log(Product_{j>=1} (1 - j^k*x^j)^(1/j)).

%F A(n,k) = Sum_{d|n} (n/d)^(k*d).

%e Square array begins:

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

%e 2, 3, 5, 9, 17, 33, 65, ...

%e 2, 4, 10, 28, 82, 244, 730, ...

%e 3, 9, 33, 129, 513, 2049, 8193, ...

%e 2, 6, 26, 126, 626, 3126, 15626, ...

%e 4, 24, 182, 1458, 11954, 99594, 840242, ...

%t T[n_, k_] := DivisorSum[n, #^(k*n/#) &]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 11 2021 *)

%o (PARI) T(n,k) = sumdiv(n, d, (n/d)^(k*d));

%o matrix(9, 9, n, k, T(n,k-1)) \\ _Michel Marcus_, Jun 02 2019

%Y Columns k=0..3 give A000005, A055225, A073705, A073706.

%Y Cf. A294579.

%K nonn,tabl

%O 1,3

%A _Seiichi Manyama_, Jun 02 2019