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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)).
5
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.
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
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Jul 26 2017
STATUS
approved