A308509 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n/d).

1, 1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 10, 9, 2, 1, 17, 28, 33, 6, 4, 1, 33, 82, 129, 26, 24, 2, 1, 65, 244, 513, 126, 182, 8, 4, 1, 129, 730, 2049, 626, 1458, 50, 41, 3, 1, 257, 2188, 8193, 3126, 11954, 344, 577, 37, 4, 1, 513, 6562, 32769, 15626, 99594, 2402, 8705, 811, 68, 2

Offset

1,3

Links

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Formula

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

A(n,k) = Sum_{d|n} (n/d)^(k*d).

Example

Square array begins:
   1,  1,   1,    1,     1,     1,      1, ...
   2,  3,   5,    9,    17,    33,     65, ...
   2,  4,  10,   28,    82,   244,    730, ...
   3,  9,  33,  129,   513,  2049,   8193, ...
   2,  6,  26,  126,   626,  3126,  15626, ...
   4, 24, 182, 1458, 11954, 99594, 840242, ...

Mathematica

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

Prog

(PARI) T(n, k) = sumdiv(n, d, (n/d)^(k*d));
matrix(9, 9, n, k, T(n, k-1)) \\ Michel Marcus, Jun 02 2019

Crossrefs

Columns k=0..3 give A000005, A055225, A073705, A073706.

Cf. A294579.

Sequence in context: A129262 A322263 A279394 * A280514 A246105 A211980

Adjacent sequences: A308506 A308507 A308508 * A308510 A308511 A308512

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 02 2019

Status

approved