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A319753
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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)/(1 - k*x^j).
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2
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1, 1, -1, 1, 0, -1, 1, 1, 0, 0, 1, 2, 3, 0, 0, 1, 3, 8, 6, 0, 1, 1, 4, 15, 24, 14, 0, 0, 1, 5, 24, 60, 78, 27, 0, 1, 1, 6, 35, 120, 252, 232, 60, 0, 0, 1, 7, 48, 210, 620, 1005, 720, 117, 0, 0, 1, 8, 63, 336, 1290, 3096, 4080, 2152, 246, 0, 0, 1, 9, 80, 504, 2394, 7735, 15600, 16305, 6528, 490, 0, 0
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OFFSET
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0,12
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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)/(1 - k*x^j).
G.f. of column k: exp(Sum_{j>=1} ( Sum_{d|j} d*(k^(j/d) - 1) ) * x^j/j).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
-1, 0, 1, 2, 3, 4, ...
-1, 0, 3, 8, 15, 24, ...
0, 0, 6, 24, 60, 120, ...
0, 0, 14, 78, 252, 620, ...
1, 0, 27, 232, 1005, 3096, ...
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MATHEMATICA
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Table[Function[k, SeriesCoefficient[Product[(1 - x^i)/(1 - k x^i), {i, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Exp[Sum[Sum[d (k^(i/d) - 1), {d, Divisors[i]}] x^i/i, {i, n}]], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
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CROSSREFS
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Columns k=0..20 give A010815, A000007, A006951, A006952, A049314, A049315, A221578, A049316, A182603, A182604, A221579, A182605, A221580, A182606, A221581, A221582, A182607, A182608, A221583, A182609, A221584.
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KEYWORD
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AUTHOR
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STATUS
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approved
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