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).

1, 1, 3, 1, 9, 4, 1, 33, 82, 7, 1, 129, 2188, 1033, 6, 1, 513, 59050, 262177, 15626, 12, 1, 2049, 1594324, 67108993, 48828126, 280026, 8, 1, 8193, 43046722, 17179869697, 152587890626, 13060696236, 5764802, 15, 1, 32769, 1162261468, 4398046513153, 476837158203126, 609359740069674, 4747561509944, 134218761, 13

Offset: 1

Author

Seiichi Manyama, Jun 18 2019

Keywords

• nonn
• tabl

Examples

			Square array begins:
   1,     1,        1,            1,               1, ...
   3,     9,       33,          129,             513, ...
   4,    82,     2188,        59050,         1594324, ...
   7,  1033,   262177,     67108993,     17179869697, ...
   6, 15626, 48828126, 152587890626, 476837158203126, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 1..52, flattened

Crossrefs

Columns k=0..3 give A000203, A283498, A283533, A283535.

Row n=1..2 give A000012, A087289.

Cf. A294579, A308698, A308701.

Programs

Mathematica

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

Formula

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

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