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A308704 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d+1).

%I A308704 #30 May 09 2021 02:50:25
%S A308704 1,1,3,1,9,4,1,33,82,7,1,129,2188,1033,6,1,513,59050,262177,15626,12,
%T A308704 1,2049,1594324,67108993,48828126,280026,8,1,8193,43046722,
%U A308704 17179869697,152587890626,13060696236,5764802,15,1,32769,1162261468,4398046513153,476837158203126,609359740069674,4747561509944,134218761,13
%N A308704 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d+1).
%H A308704 Seiichi Manyama, <a href="/A308704/b308704.txt">Antidiagonals n = 1..52, flattened</a>
%F A308704 L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^(j^(k*j))).
%F A308704 G.f. of column k: Sum_{j>=1} j^(k*j+1) * x^j/(1 - x^j).
%e A308704 Square array begins:
%e A308704    1,     1,        1,            1,               1, ...
%e A308704    3,     9,       33,          129,             513, ...
%e A308704    4,    82,     2188,        59050,         1594324, ...
%e A308704    7,  1033,   262177,     67108993,     17179869697, ...
%e A308704    6, 15626, 48828126, 152587890626, 476837158203126, ...
%t A308704 T[n_, k_] := DivisorSum[n, #^(k*# + 1) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 09 2021 *)
%Y A308704 Columns k=0..3 give A000203, A283498, A283533, A283535.
%Y A308704 Row n=1..2 give A000012, A087289.
%Y A308704 Cf. A294579, A308698, A308701.
%K A308704 nonn,tabl
%O A308704 1,3
%A A308704 _Seiichi Manyama_, Jun 18 2019