A308569 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n).

1, 1, 2, 1, 5, 2, 1, 17, 28, 3, 1, 65, 730, 273, 2, 1, 257, 19684, 65793, 3126, 4, 1, 1025, 531442, 16781313, 9765626, 47450, 2, 1, 4097, 14348908, 4295032833, 30517578126, 2177317874, 823544, 4, 1, 16385, 387420490, 1099512676353, 95367431640626, 101560344351050, 678223072850, 16843009, 3

Offset: 1

Author

Seiichi Manyama, Jun 08 2019

Keywords

• nonn
• tabl

Examples

			Square array begins:
   1,    1,       1,           1,              1, ...
   2,    5,      17,          65,            257, ...
   2,   28,     730,       19684,         531442, ...
   3,  273,   65793,    16781313,     4295032833, ...
   2, 3126, 9765626, 30517578126, 95367431640626, ...
		

Links

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

Crossrefs

Columns k=0..2 give A000005, A023887, A308570.

Rows n=1..2 give A000012, A052539.

A(n,n) gives A308571.

Cf. A308504, A308509.

Programs

Mathematica

T[n_, k_] := DivisorSum[n, #^(k*n) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

PARI

T(n,k) = sumdiv(n, d, d^(k*n));
matrix(5, 5, n, k, T(n,k-1)) \\ Michel Marcus, Jun 08 2019

Formula

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