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A308698 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d).

%I A308698 #31 May 09 2021 02:50:47
%S A308698 1,1,2,1,5,2,1,17,28,3,1,65,730,261,2,1,257,19684,65553,3126,4,1,1025,
%T A308698 531442,16777281,9765626,46688,2,1,4097,14348908,4294967553,
%U A308698 30517578126,2176783082,823544,4,1,16385,387420490,1099511628801,95367431640626,101559956688164,678223072850,16777477,3
%N A308698 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d).
%H A308698 Seiichi Manyama, <a href="/A308698/b308698.txt">Antidiagonals n = 1..52, flattened</a>
%F A308698 L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^(j^(k*j-1))).
%F A308698 G.f. of column k: Sum_{j>=1} j^(k*j) * x^j/(1 - x^j).
%e A308698 Square array begins:
%e A308698    1,    1,       1,           1,              1, ...
%e A308698    2,    5,      17,          65,            257, ...
%e A308698    2,   28,     730,       19684,         531442, ...
%e A308698    3,  261,   65553,    16777281,     4294967553, ...
%e A308698    2, 3126, 9765626, 30517578126, 95367431640626, ...
%t A308698 T[n_, k_] := DivisorSum[n, #^(k*#) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 09 2021 *)
%Y A308698 Columns k=0..3 give A000005, A062796, A308696, A308697.
%Y A308698 Row n=1..2 give A000012, A052539.
%Y A308698 Cf. A308509, A308701, A308704.
%K A308698 nonn,tabl
%O A308698 1,3
%A A308698 _Seiichi Manyama_, Jun 17 2019