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EXAMPLE
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a(0) = 1;
a(1) = [x^1] 1/(1 + x) = -1;
a(2) = [x^2] 1/((1 + 2*x)*(1 + x^2)) = 3;
a(3) = [x^3] 1/((1 + 3*x)*(1 + 2*x^2)*(1 + x^3)) = -22;
a(4) = [x^4] 1/((1 + 4*x)*(1 + 3*x^2)*(1 + 2*x^3)*(1 + x^4)) = 224;
a(5) = [x^5] 1/((1 + 5*x)*(1 + 4*x^2)*(1 + 3*x^3)*(1 + 2*x^4)*(1 + x^5)) = -2759, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 + (n - k + 1)*x^k) begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (-1), 1, -1, 1, -1, ...
n = 2: 1, -2, (3), -6, 13, -26, ...
n = 3: 1, -3, 7, (-22), 70, -208, ...
n = 4: 1, -4, 13, -54, (224), -890, ...
n = 5: 1, -5, 21, -108, 554, (-2759), ...
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MATHEMATICA
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Table[SeriesCoefficient[Product[1/(1 + (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]
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