OFFSET
1,4
COMMENTS
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10010
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)
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