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A327315
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Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (x-2)/(x^2-x+1)).
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2
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-2, 1, -1, 4, -1, 1, 3, -6, 1, 2, -4, -6, 8, -1, 1, -10, 10, 10, -10, 1, -1, -6, 30, -20, -15, 12, -1, -2, 7, 21, -70, 35, 21, -14, 1, -1, 16, -28, -56, 140, -56, -28, 16, -1, 1, 9, -72, 84, 126, -252, 84, 36, -18, 1, 2, -10, -45, 240, -210, -252, 420, -120
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OFFSET
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0,1
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COMMENTS
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Conjecture: The numbers n for which the n-th polynomial is irreducible are given by A069353.
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LINKS
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EXAMPLE
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First eight rows:
-2, 1;
-1, 4, -1;
1, 3, -6, 1;
2, -4, -6, 8, -1;
1, -10, 10, 10, -10, 1;
-1, -6, 30, -20, -15, 12, -1;
-2, 7, 21, -70, 35, 21, -14, 1;
-1, 16, -28, -56, 140, -56, -28, 16, -1;
First eight polynomials:
-2 + x
-1 + 4 x - x^2
1 + 3 x - 6 x^2 + x^3
2 - 4 x - 6 x^2 + 8 x^3 - x^4
(1 + x) (1 - 11 x + 21 x^2 - 11 x^3 + x^4)
-1 - 6 x + 30 x^2 - 20 x^3 - 15 x^4 + 12 x^5 - x^6
(-2 + x) (1 - 3 x - 12 x^2 + 29 x^3 - 3 x^4 - 12 x^5 + x^6)
-1 + 16 x - 28 x^2 - 56 x^3 + 140 x^4 - 56 x^5 - 28 x^6 + 16 x^7 - x^8
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MATHEMATICA
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g[x_, n_] := Numerator[ Factor[D[(x - 2)/(x^2 - x + 1), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* A327315 polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x];
Table[h[n], {n, 0, 10}] (* A327315 sequence *)
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CROSSREFS
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KEYWORD
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tabf,sign
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AUTHOR
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STATUS
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approved
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