A322103 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} sigma_k(d).

1, 1, 3, 1, 4, 3, 1, 6, 5, 6, 1, 10, 11, 11, 3, 1, 18, 29, 27, 7, 9, 1, 34, 83, 83, 27, 20, 3, 1, 66, 245, 291, 127, 66, 9, 10, 1, 130, 731, 1091, 627, 290, 51, 26, 6, 1, 258, 2189, 4227, 3127, 1494, 345, 112, 18, 9, 1, 514, 6563, 16643, 15627, 8330, 2403, 668, 102, 28, 3

Offset

1,3

Links

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

Formula

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

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

Example

Square array begins:
  1,   1,   1,    1,     1,     1,  ...
  3,   4,   6,   10,    18,    34,  ...
  3,   5,  11,   29,    83,   245,  ...
  6,  11,  27,   83,   291,  1091,  ...
  3,   7,  27,  127,   627,  3127,  ...
  9,  20,  66,  290,  1494,  8330,  ...

Mathematica

Table[Function[k, Sum[DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[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*numdiv(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 A007425, A007429, A007433, A321140.

Cf. A109974, A321141 (diagonal), A356045.

Sequence in context: A104568 A030758 A347065 * A272172 A104764 A152842

Adjacent sequences: A322100 A322101 A322102 * A322104 A322105 A322106

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Nov 26 2018

Status

approved