A322104 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).

1, 1, 5, 1, 7, 7, 1, 11, 13, 17, 1, 19, 31, 35, 11, 1, 35, 85, 95, 31, 35, 1, 67, 247, 311, 131, 91, 15, 1, 131, 733, 1127, 631, 341, 57, 49, 1, 259, 2191, 4295, 3131, 1615, 351, 155, 34, 1, 515, 6565, 16775, 15631, 8645, 2409, 775, 130, 55, 1, 1027, 19687, 66311, 78131, 49111, 16815, 4991, 850, 217, 23

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Formula

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

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

A(n,k) = Sum_{d|n} d^(k+1)*sigma_1(n/d).

Example

Square array begins:
   1,   1,    1,     1,     1,      1,  ...
   5,   7,   11,    19,    35,     67,  ...
   7,  13,   31,    85,   247,    733,  ...
  17,  35,   95,   311,  1127,   4295,  ...
  11,  31,  131,   631,  3131,  15631,  ...
  35,  91,  341,  1615,  8645,  49111,  ...

Mathematica

Table[Function[k, Sum[d DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

Prog

(PARI) T(n, k)={sumdiv(n, d, d^(k+1)*sigma(n/d))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

Crossrefs

Columns k=0..3 give A060640, A001001, A027847, A027848.

Cf. A109974, A320940 (diagonal), A321876, A322103.

Sequence in context: A187561 A088515 A200638 * A100122 A368668 A001945

Adjacent sequences: A322101 A322102 A322103 * A322105 A322106 A322107

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Nov 26 2018

Status

approved