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A286335
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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 + x^j)^k.
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35
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1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 2, 0, 1, 5, 10, 13, 9, 3, 0, 1, 6, 15, 24, 24, 14, 4, 0, 1, 7, 21, 40, 51, 42, 22, 5, 0, 1, 8, 28, 62, 95, 100, 73, 32, 6, 0, 1, 9, 36, 91, 162, 206, 190, 120, 46, 8, 0, 1, 10, 45, 128, 259, 384, 425, 344, 192, 66, 10, 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 (or odd parts) with k types 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^j)^k.
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EXAMPLE
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A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part).
Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part).
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 1, 3, 6, 10, 15, ...
0, 2, 6, 13, 24, 40, ...
0, 2, 9, 24, 51, 95, ...
0, 3, 14, 42, 100, 206, ...
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
(t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..n/i)))
end:
A:= (n, k)-> b(n$2, k):
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MATHEMATICA
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Table[Function[k, SeriesCoefficient[Product[(1 + x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 11}, {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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