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A249303
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Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
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1
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1, 0, 1, -1, 2, -1, 1, 1, 0, -2, 3, 1, -4, 3, 1, 1, -2, -2, 4, 0, 3, -9, 6, 1, -1, 6, -9, 0, 5, -1, 3, 3, -15, 10, 1, 0, -4, 18, -24, 5, 6, 1, -8, 18, -6, -20, 15, 1, 1, -4, -4, 36, -49, 14, 7, 0, 5, -30, 60, -35, -21, 21, 1, -1, 10, -30, 20, 50, -84, 28, 8
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OFFSET
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0,5
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COMMENTS
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The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = 1 + (x - 1)/f(n-1,x), where f(0,x) = 1.
Every row sum is 1. The first column is purely periodic with period (1,0,-1,-1,0,1).
Conjecture: for n > 2, p(n,x) is irreducible if and only if n is a (prime - 2). More generally, if c is arbitrary and f(n,x) = 1 + (x + c)/f(n-1,x), where f(x,0) = 1, then p(n,x) is irreducible if and only if n is a (prime - 2).
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LINKS
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Clark Kimberling, Rows 0..100, flattened
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EXAMPLE
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f(0,x) = 1/1, so that p(0,x) = 1
f(1,x) = x/1, so that p(1,x) = x;
f(2,x) = (-1 + 2 x)/x, so that p(2,x) = -1 + 2 x.
First 6 rows of the triangle of coefficients:
... 1
... 0 ... 1
.. -1 ... 2
.. -1 ... 1 ... 1
... 0 .. -2 ... 3
... 1 .. -4 ... 3 ... 1
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MATHEMATICA
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z = 20; f[n_, x_] := 1 + (x - 1)/f[n - 1, x]; f[0, x_] = 1;
t = Table[Factor[f[n, x]], {n, 0, z}]
u = Numerator[t]
TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]] (*A249303 array*)
v = Flatten[CoefficientList[u, x]] (*A249303*)
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CROSSREFS
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Cf. A128100, A229057, A229074.
Sequence in context: A281743 A118404 A089339 * A319081 A182662 A308778
Adjacent sequences: A249300 A249301 A249302 * A249304 A249305 A249306
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KEYWORD
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tabf,sign,easy
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AUTHOR
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Clark Kimberling, Oct 24 2014
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STATUS
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approved
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