A308698 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, 2, 1, 5, 2, 1, 17, 28, 3, 1, 65, 730, 261, 2, 1, 257, 19684, 65553, 3126, 4, 1, 1025, 531442, 16777281, 9765626, 46688, 2, 1, 4097, 14348908, 4294967553, 30517578126, 2176783082, 823544, 4, 1, 16385, 387420490, 1099511628801, 95367431640626, 101559956688164, 678223072850, 16777477, 3

Offset

1,3

Links

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

Formula

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

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

Example

Square array begins:
   1,    1,       1,           1,              1, ...
   2,    5,      17,          65,            257, ...
   2,   28,     730,       19684,         531442, ...
   3,  261,   65553,    16777281,     4294967553, ...
   2, 3126, 9765626, 30517578126, 95367431640626, ...

Mathematica

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

Crossrefs

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

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

Cf. A308509, A308701, A308704.

Sequence in context: A275431 A281348 A371305 * A308569 A350073 A174978

Adjacent sequences: A308695 A308696 A308697 * A308699 A308700 A308701

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 17 2019

Status

approved