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A308701 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 A308701 #37 May 09 2021 02:50:40
%S A308701 1,1,2,1,3,2,1,5,10,3,1,9,82,67,2,1,17,730,4101,626,4,1,33,6562,
%T A308701 262153,390626,7788,2,1,65,59050,16777233,244140626,60466262,117650,4,
%U A308701 1,129,531442,1073741857,152587890626,470184985314,13841287202,2097219,3
%N A308701 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 A308701 Seiichi Manyama, <a href="/A308701/b308701.txt">Antidiagonals n = 1..53, flattened</a>
%F A308701 L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^(j^(k*j-k-1))).
%F A308701 G.f. of column k: Sum_{j>=1} j^(k*(j-1)) * x^j/(1 - x^j).
%e A308701 Square array begins:
%e A308701    1,   1,      1,         1,            1, ...
%e A308701    2,   3,      5,         9,           17, ...
%e A308701    2,  10,     82,       730,         6562, ...
%e A308701    3,  67,   4101,    262153,     16777233, ...
%e A308701    2, 626, 390626, 244140626, 152587890626, ...
%t A308701 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 A308701 Columns k=0..2 give A000005, A262843, A308753.
%Y A308701 Row n=1..3 give A000012, A000051, A062396.
%Y A308701 Cf. A308694, A308698, A308704.
%K A308701 nonn,tabl
%O A308701 1,3
%A A308701 _Seiichi Manyama_, Jun 22 2019