A328611 Irregular triangular array read by rows: row n gives the coefficients of the second of two factors of even-degree polynomials described in Comments.

0, -1, 1, 4, -1, 3, 3, 6, -1, 4, 12, 6, 8, -1, 7, 20, 30, 10, 10, -1, 11, 42, 60, 60, 15, 12, -1, 18, 77, 147, 140, 105, 21, 14, -1, 29, 144, 308, 392, 280, 168, 28, 16, -1, 47, 261, 648, 924, 882, 504, 252, 36, 18, -1, 76, 470, 1305, 2160, 2310, 1764, 840

Offset: 1

Author

Clark Kimberling, Oct 24 2019

Keywords

• tabf
• sign

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 Lucas numbers, L(k), for k >= 1; see A000032. It appears that after the first row, the row sums are L(2k+1), for k >= 1.

Examples

			First nine rows:
.
   0,  -1; (coefficients of -x)
   1,   4,  -1; (coefficients of 1 + 4*x - x^2)
   3,   3,   6,  -1;
   4,  12,   6,   8,  -1;
   7,  20,  30,  10,  10,  -1;
  11,  42,  60,  60,  15,  12,  -1;
  18,  77, 147, 140, 105,  21,  14, -1;
  29, 144, 308, 392, 280, 168,  28, 16, -1;
  47, 261, 648, 924, 882, 504, 252, 36, 18, -1;
		

Links

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

Crossrefs

Cf. A000032, A326925, A328610.

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 *)