A308504 - OEIS

%I #34 May 11 2021 01:55:10

%S 1,1,5,1,9,28,1,17,82,273,1,33,244,1057,3126,1,65,730,4161,15626,

%T 47450,1,129,2188,16513,78126,282252,823544,1,257,6562,65793,390626,

%U 1686434,5764802,16843009,1,513,19684,262657,1953126,10097892,40353608,134480385,387440173

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

%H Seiichi Manyama, <a href="/A308504/b308504.txt">Antidiagonals n = 1..140, flattened</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F L.g.f. of column k: -log(Product_{j>=1} (1 - (j*x)^j)^(j^(k-1))).

%F a((i-1)*i/2 + j) = sigma_i(j) for 1 <= j <= i.

%e a(4) = a(2*3/2 + 1) = sigma_3(1) = 1.

%e a(5) = a(2*3/2 + 2) = sigma_3(2) = 1^3 + 2^3 = 9.

%e a(6) = a(2*3/2 + 3) = sigma_3(3) = 1^3 + 3^3 = 28.

%e Square array begins:

%e        1,      1,       1,        1,        1, ...

%e        5,      9,      17,       33,       65, ...

%e       28,     82,     244,      730,     2188, ...

%e      273,   1057,    4161,    16513,    65793, ...

%e     3126,  15626,   78126,   390626,  1953126, ...

%e    47450, 282252, 1686434, 10097892, 60526250, ...

%t 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 Columns k=0..2 give A023887, A294645, A294810.

%Y A(n,n) gives A308570.

%Y Cf. A279394, A308502.

%K nonn,tabl

%O 1,3

%A _Seiichi Manyama_, Jun 02 2019