A321876 -id:A321876 - cp's OEIS frontend

A321877 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^sigma_k(j).

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 7, 6, 1, 1, 9, 15, 14, 10, 1, 1, 17, 37, 41, 28, 17, 1, 1, 33, 99, 137, 107, 58, 25, 1, 1, 65, 277, 491, 487, 286, 106, 38, 1, 1, 129, 795, 1829, 2429, 1749, 700, 201, 59, 1, 1, 257, 2317, 6971, 12763, 12056, 5901, 1735, 372, 86

Offset: 0

Ilya Gutkovskiy, Nov 20 2018

			Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   1,   1,    1,    1,     1,      1,  ...
   2,   3,    5,    9,    17,     33,  ...
   4,   7,   15,   37,    99,    277,  ...
   6,  14,   41,  137,   491,   1829,  ...
  10,  28,  107,  487,  2429,  12763,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..9 give A107742, A192065, A288414, A288415, A301548, A301549, A301550, A301551, A301552, A301553.
Main diagonal gives A321042.
Cf. A321876.

Table[Function[k, SeriesCoefficient[Product[(1 + x^j)^DivisorSigma[k, j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten
Table[Function[k, SeriesCoefficient[Exp[Sum[DivisorSigma[k + 1, j] x^j/(j (1 - x^(2 j))), {j, 1, n}]], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten

G.f. of column k: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^k).

G.f. of column k: exp(Sum_{j>=1} sigma_(k+1)(j)*x^j/(j*(1 - x^(2*j)))).

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Nov 26 2018

			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,  ...

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

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

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

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

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

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).

cp's OEIS Frontend

A321877 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^sigma_k(j).

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