OFFSET
1,1
COMMENTS
Let p(n) denote the polynomial (1/n!)*(numerator of n-th derivative of (1-x)/(1-x-x^2)). It is conjectured in A326925 that if n = 2k, then p(n) = f(k)*g(k), where f(k) and g(k) are polynomials of degree k. Row k of the present array shows the coefficients of f(k).
It appears that, after the first term, column 1 consists of the Fibonacci numbers, F(k), for k >= 1; see A000045. It appears that after the first row, the row sums are F(2k+1), and the alternating row sums are (-1)^k F(k).
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..1325
EXAMPLE
First nine rows:
.
-2, 1; (coefficients of -2 + x)
1, 0, 1; (coefficients of 1 + x^2)
1, 3, 0, 1;
2, 4, 6, 0, 1;
3, 10, 10, 10, 0, 1;
5, 18, 30, 20, 15, 0, 1;
8, 35, 63, 70, 35, 21, 0, 1;
13, 64, 140, 168, 140, 56, 28, 0, 1;
21, 117, 288, 420, 378, 252, 84, 36, 0, 1;
MATHEMATICA
g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]];
f = Table[FactorList[g[x, n]/n!], {n, 1, 60, 2}]; (* polynomials *)
r[n_] := Rest[f[[n]]];
Column[Table[First[CoefficientList[r[n][[1]], x]], {n, 1, 16}]] (* A328610 *)
Column[Table[-First[CoefficientList[r[n][[2]], x]], {n, 1, 16}]] (* A328611 *)
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Clark Kimberling, Oct 24 2019
STATUS
approved