A328610 Irregular triangular array read by rows: the rows show the coefficients of the first of two factors of even-degree polynomials described in Comments.

-2, 1, 1, 0, 1, 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, 34, 210, 585, 960, 1050, 756, 420, 120, 45, 0, 1, 55, 374

Offset: 1

Author

Clark Kimberling, Oct 24 2019

Keywords

• sign
• tabf

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).

Examples

			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;
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1325

Crossrefs

Cf. A000045, A326925, A328611.

Programs

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 *)