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A308504 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n+k).

%I A308504 #34 May 11 2021 01:55:10
%S A308504 1,1,5,1,9,28,1,17,82,273,1,33,244,1057,3126,1,65,730,4161,15626,
%T A308504 47450,1,129,2188,16513,78126,282252,823544,1,257,6562,65793,390626,
%U A308504 1686434,5764802,16843009,1,513,19684,262657,1953126,10097892,40353608,134480385,387440173
%N A308504 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n+k).
%H A308504 Seiichi Manyama, <a href="/A308504/b308504.txt">Antidiagonals n = 1..140, flattened</a>
%H A308504 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A308504 L.g.f. of column k: -log(Product_{j>=1} (1 - (j*x)^j)^(j^(k-1))).
%F A308504 a((i-1)*i/2 + j) = sigma_i(j) for 1 <= j <= i.
%e A308504 a(4) = a(2*3/2 + 1) = sigma_3(1) = 1.
%e A308504 a(5) = a(2*3/2 + 2) = sigma_3(2) = 1^3 + 2^3 = 9.
%e A308504 a(6) = a(2*3/2 + 3) = sigma_3(3) = 1^3 + 3^3 = 28.
%e A308504 Square array begins:
%e A308504        1,      1,       1,        1,        1, ...
%e A308504        5,      9,      17,       33,       65, ...
%e A308504       28,     82,     244,      730,     2188, ...
%e A308504      273,   1057,    4161,    16513,    65793, ...
%e A308504     3126,  15626,   78126,   390626,  1953126, ...
%e A308504    47450, 282252, 1686434, 10097892, 60526250, ...
%t A308504 T[n_, k_] := DivisorSum[n, #^(n+k) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 11 2021 *)
%Y A308504 Columns k=0..2 give A023887, A294645, A294810.
%Y A308504 A(n,n) gives A308570.
%Y A308504 Cf. A279394, A308502.
%K A308504 nonn,tabl
%O A308504 1,3
%A A308504 _Seiichi Manyama_, Jun 02 2019