A328649 Irregular triangular array read by rows: row n shows the coefficients of the following polynomial of degree n: (1/n!)*(numerator of n-th derivative of (x-2)/(1-x-x^2)).

2, -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

Offset

0,1

Comments

The first 201 polynomials are irreducible. Column 1: signed Lucas numbers (A000032).

Links

Table of n, a(n) for n=0..56.

Example

First eight rows:
    2,   -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;
First eight polynomials:
   2 - x
  -1 - 4 x + x^2
   3 + 3 x + 6 x^2 - x^3
  -4 - 12 x - 6 x^2 - 8 x^3 + x^4
   7 + 20 x + 30 x^2 + 10 x^3 + 10 x^4 - x^5
  -11 - 42 x - 60 x^2 - 60 x^3 - 15 x^4 - 12 x^5 + x^6
   18 + 77 x + 147 x^2 + 140 x^3 + 105 x^4 + 21 x^5 + 14 x^6 - x^7
  -29 - 144 x - 308 x^2 - 392 x^3 - 280 x^4 - 168 x^5 - 28 x^6 - 16 x^7 + x^8

Mathematica

g[x_, n_] := Numerator[ Factor[D[(x - 2)/(1 - x - x^2), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]]  (* A328649 polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x];
Table[h[n], {n, 0, 10}] (* A328649 sequence *)
Column[%] (* A328649 array *)

Crossrefs

Cf. A000032, A328646.

Sequence in context: A249061 A334178 A210209 * A281422 A344529 A144389

Adjacent sequences: A328646 A328647 A328648 * A328650 A328651 A328652

Keyword

tabf,sign

Author

Clark Kimberling, Nov 01 2019

Status

approved