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A308569 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n).

%I A308569 #27 May 11 2021 01:55:00
%S A308569 1,1,2,1,5,2,1,17,28,3,1,65,730,273,2,1,257,19684,65793,3126,4,1,1025,
%T A308569 531442,16781313,9765626,47450,2,1,4097,14348908,4295032833,
%U A308569 30517578126,2177317874,823544,4,1,16385,387420490,1099512676353,95367431640626,101560344351050,678223072850,16843009,3
%N A308569 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n).
%H A308569 Seiichi Manyama, <a href="/A308569/b308569.txt">Antidiagonals n = 1..52, flattened</a>
%F A308569 L.g.f. of column k: -log(Product_{j>=1} (1 - (j^k*x)^j)^(1/j)).
%e A308569 Square array begins:
%e A308569    1,    1,       1,           1,              1, ...
%e A308569    2,    5,      17,          65,            257, ...
%e A308569    2,   28,     730,       19684,         531442, ...
%e A308569    3,  273,   65793,    16781313,     4295032833, ...
%e A308569    2, 3126, 9765626, 30517578126, 95367431640626, ...
%t A308569 T[n_, k_] := DivisorSum[n, #^(k*n) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 11 2021 *)
%o A308569 (PARI) T(n,k) = sumdiv(n, d, d^(k*n));
%o A308569 matrix(5, 5, n, k, T(n,k-1)) \\ _Michel Marcus_, Jun 08 2019
%Y A308569 Columns k=0..2 give A000005, A023887, A308570.
%Y A308569 Rows n=1..2 give A000012, A052539.
%Y A308569 A(n,n) gives A308571.
%Y A308569 Cf. A308504, A308509.
%K A308569 nonn,tabl
%O A308569 1,3
%A A308569 _Seiichi Manyama_, Jun 08 2019