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A308509 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n/d).

%I A308509 #26 May 11 2021 01:55:03
%S A308509 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 A308509 65,244,513,126,182,8,4,1,129,730,2049,626,1458,50,41,3,1,257,2188,
%U A308509 8193,3126,11954,344,577,37,4,1,513,6562,32769,15626,99594,2402,8705,811,68,2
%N A308509 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 A308509 Seiichi Manyama, <a href="/A308509/b308509.txt">Antidiagonals n = 1..140, flattened</a>
%F A308509 L.g.f. of column k: -log(Product_{j>=1} (1 - j^k*x^j)^(1/j)).
%F A308509 A(n,k) = Sum_{d|n} (n/d)^(k*d).
%e A308509 Square array begins:
%e A308509    1,  1,   1,    1,     1,     1,      1, ...
%e A308509    2,  3,   5,    9,    17,    33,     65, ...
%e A308509    2,  4,  10,   28,    82,   244,    730, ...
%e A308509    3,  9,  33,  129,   513,  2049,   8193, ...
%e A308509    2,  6,  26,  126,   626,  3126,  15626, ...
%e A308509    4, 24, 182, 1458, 11954, 99594, 840242, ...
%t A308509 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 A308509 (PARI) T(n,k) = sumdiv(n, d, (n/d)^(k*d));
%o A308509 matrix(9, 9, n, k, T(n,k-1)) \\ _Michel Marcus_, Jun 02 2019
%Y A308509 Columns k=0..3 give A000005, A055225, A073705, A073706.
%Y A308509 Cf. A294579.
%K A308509 nonn,tabl
%O A308509 1,3
%A A308509 _Seiichi Manyama_, Jun 02 2019