A249303 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

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

Offset: 0

Author

Clark Kimberling, Oct 24 2014

Keywords

• tabf
• sign
• easy

Comments

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).

Examples

			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
		

Links

• Clark Kimberling, Rows 0..100, flattened

Crossrefs

Cf. A128100, A229057, A229074.

Programs

Mathematica

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 *)