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)).

1, 1, 2, 1, 3, 2, 1, 5, 10, 3, 1, 9, 82, 67, 2, 1, 17, 730, 4101, 626, 4, 1, 33, 6562, 262153, 390626, 7788, 2, 1, 65, 59050, 16777233, 244140626, 60466262, 117650, 4, 1, 129, 531442, 1073741857, 152587890626, 470184985314, 13841287202, 2097219, 3

Offset

1,3

Links

Seiichi Manyama, Antidiagonals n = 1..53, flattened

Formula

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

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

Example

Square array begins:
   1,   1,      1,         1,            1, ...
   2,   3,      5,         9,           17, ...
   2,  10,     82,       730,         6562, ...
   3,  67,   4101,    262153,     16777233, ...
   2, 626, 390626, 244140626, 152587890626, ...

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 *)

Crossrefs

Columns k=0..2 give A000005, A262843, A308753.

Row n=1..3 give A000012, A000051, A062396.

Cf. A308694, A308698, A308704.

Sequence in context: A119442 A064861 A305299 * A191528 A191788 A070979

Adjacent sequences: A308698 A308699 A308700 * A308702 A308703 A308704

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 22 2019

Status

approved