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Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d).
5

%I #31 May 09 2021 02:50:47

%S 1,1,2,1,5,2,1,17,28,3,1,65,730,261,2,1,257,19684,65553,3126,4,1,1025,

%T 531442,16777281,9765626,46688,2,1,4097,14348908,4294967553,

%U 30517578126,2176783082,823544,4,1,16385,387420490,1099511628801,95367431640626,101559956688164,678223072850,16777477,3

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

%H Seiichi Manyama, <a href="/A308698/b308698.txt">Antidiagonals n = 1..52, flattened</a>

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

%F G.f. of column k: Sum_{j>=1} j^(k*j) * x^j/(1 - x^j).

%e Square array begins:

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

%e 2, 5, 17, 65, 257, ...

%e 2, 28, 730, 19684, 531442, ...

%e 3, 261, 65553, 16777281, 4294967553, ...

%e 2, 3126, 9765626, 30517578126, 95367431640626, ...

%t T[n_, k_] := DivisorSum[n, #^(k*#) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 09 2021 *)

%Y Columns k=0..3 give A000005, A062796, A308696, A308697.

%Y Row n=1..2 give A000012, A052539.

%Y Cf. A308509, A308701, A308704.

%K nonn,tabl

%O 1,3

%A _Seiichi Manyama_, Jun 17 2019