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URL: https://oeis.org/A326925

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A326925
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 (1-x)/(1-x-x^2)).
6
1, -1, 0, 2, -1, 1, 0, 3, -1, 1, 4, 0, 4, -1, 2, 5, 10, 0, 5, -1, 3, 12, 15, 20, 0, 6, -1, 5, 21, 42, 35, 35, 0, 7, -1, 8, 40, 84, 112, 70, 56, 0, 8, -1, 13, 72, 180, 252, 252, 126, 84, 0, 9, -1, 21, 130, 360, 600, 630, 504, 210, 120, 0, 10, -1, 34, 231, 715
OFFSET
1,4
COMMENTS
Column 1: Fibonacci numbers, F(m), for m >= -1, as in A000045. For n >= 0, the n-th row sum = F(2n), as in A001906.
Conjecture: The odd degree polynomials are irreducible; the even degree (= 2k) polynomials have exactly two irreducible factors, each of degree k.
LINKS
FORMULA
G.f. as array: ((y^2 + y - 1)*x - y + 1)/(1 + (y^2 + y - 1)*x^2 + (-2*y - 1)*x). - Robert Israel, Oct 31 2019
EXAMPLE
First 7 rows:
1 -1
0 2 -1
1 0 3 -1
1 4 0 4 -1
2 5 0 10 5 -1
3 12 15 20 0 6 -1
5 21 42 35 35 0 7 -1
First 7 polynomials:
1 - x
2 x - x^2
1 + 3 x^2 - x^3
1 + 4 x + 4 x^3 - x^4
2 + 5 x + 10 x^2 + 5 x^4 - x^5
3 + 12 x + 15 x^2 + 20 x^3 + 6 x^5 - x^6
5 + 21 x + 42 x^2 + 35 x^3 + 35 x^4 + 7 x^6 - x^7
Factorizations of even-degree polynomials:
degree 2: (2 - x)*x
degree 4: (1 + x^2)*(1 + 4x - x^2)
degree 6: (1 + 3x + x^3)*(3 + 3x + 6x^2 - x^3)
degree 8: (2 + 4x + 6x^2 + x^4)*(4 + 12 x + 6x^2 + 8x^3 - x^4)
degree 10: (3 + 10 x + 10 x^2 + 10 x^3 + x^5)*(7 + 20 x + 30 x^2 + 10 x^3 + 10 x^4 - x^5)
It appears that the constant terms of the factors are Fibonacci numbers (A000045) and Lucas numbers (A000032).
MATHEMATICA
g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x] (* A326925 *)
Table[h[n], {n, 0, 10}]
Column[%]
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Clark Kimberling, Oct 22 2019
STATUS
approved