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A286957
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Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + k*x^j).
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5
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1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 2, 0, 1, 4, 3, 6, 2, 0, 1, 5, 4, 12, 6, 3, 0, 1, 6, 5, 20, 12, 10, 4, 0, 1, 7, 6, 30, 20, 21, 18, 5, 0, 1, 8, 7, 42, 30, 36, 48, 22, 6, 0, 1, 9, 8, 56, 42, 55, 100, 57, 30, 8, 0, 1, 10, 9, 72, 56, 78, 180, 116, 84, 42, 10, 0, 1, 11, 10, 90, 72, 105, 294, 205, 180, 120, 66, 12, 0
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OFFSET
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0,8
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COMMENTS
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A(n,k) is the number of partitions of n into distinct parts of k sorts: the parts are unordered, but not the sorts.
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LINKS
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FORMULA
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G.f. of column k: Product_{j>=1} (1 + k*x^j).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 1, 2, 3, 4, 5, ...
0, 2, 6, 12, 20, 30, ...
0, 2, 6, 12, 20, 30, ...
0, 3, 10, 21, 36, 55, ...
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MATHEMATICA
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Table[Function[k, SeriesCoefficient[Product[(1 + k x^i), {i, 1, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-k, x]/(1 + k), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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