A290307 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 0, 1, 1, 1, 2, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 2, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 3, 2, 0, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 4, 2, 0, 1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 5, 3, 0

Offset

0,25

Comments

A(n,k) is the number of partitions of n into distinct parts where no part is a multiple of k.

Links

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

Index entries for sequences related to partitions

Formula

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

For asymptotics of column k see comment from Vaclav Kotesovec in A261772.

Example

Square array begins:
  1,  1,  1,  1,  1,  1, ...
  0,  1,  1,  1,  1,  1, ...
  0,  0,  1,  1,  1,  1, ...
  0,  1,  1,  2,  2,  2, ...
  0,  1,  1,  1,  2,  2, ...
  0,  1,  2,  2,  2,  3, ...

Mathematica

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)/(1 + x^(i k)), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-1, x]/QPochhammer[-1, x^k], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten

Crossrefs

Columns k=1-10 give: A000007, A000700, A003105, A070048, A096938, A261770, A097793, A261771, A112193, A261772.

Cf. A286653.

Sequence in context: A037870 A250205 A326017 * A206588 A302234 A345007

Adjacent sequences: A290304 A290305 A290306 * A290308 A290309 A290310

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Jul 26 2017

Status

approved