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A286653
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Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).
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9
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1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 3, 4, 3, 0, 1, 1, 2, 3, 4, 5, 4, 0, 1, 1, 2, 3, 5, 6, 7, 5, 0, 1, 1, 2, 3, 5, 6, 9, 9, 6, 0, 1, 1, 2, 3, 5, 7, 10, 12, 13, 8, 0, 1, 1, 2, 3, 5, 7, 10, 13, 16, 16, 10, 0, 1, 1, 2, 3, 5, 7, 11, 14, 19, 22, 22, 12, 0
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OFFSET
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0,13
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COMMENTS
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A(n,k) is the number of partitions of n in which no parts are multiples of k.
A(n,k) is also the number of partitions of n into at most k-1 copies of each part.
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LINKS
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FORMULA
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G.f. of column k: Product_{j>=1} (1 - x^(k*j))/(1 - x^j).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, ...
0, 2, 2, 3, 3, 3, ...
0, 2, 4, 4, 5, 5, ...
0, 3, 5, 6, 6, 7, ...
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k*i*(i+1)/2<n, 0,
add((l->[0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
end:
A:= (n, k)-> b(n$2, k-1)[1]:
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MATHEMATICA
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Table[Function[k, SeriesCoefficient[Product[(1 - x^(i k))/(1 - x^i), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[x^k, x^k]/QPochhammer[x, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
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CROSSREFS
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Columns k=1-13 give: A000007, A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546, A341714.
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KEYWORD
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AUTHOR
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STATUS
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approved
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