A328646 Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (1-x)/(x^2-3x+1)).

1, -1, 2, -2, 1, 5, -6, 3, -1, 13, -20, 12, -4, 1, 34, -65, 50, -20, 5, -1, 89, -204, 195, -100, 30, -6, 1, 233, -623, 714, -455, 175, -42, 7, -1, 610, -1864, 2492, -1904, 910, -280, 56, -8, 1, 1597, -5490, 8388, -7476, 4284, -1638, 420, -72, 9, -1, 4181

Offset

0,3

Comments

The first 501 polynomials are irreducible. Column 1 of the array: A001519 (odd-indexed Fibonacci numbers). Row sums: A000045 (Fibonacci numbers). Alternating row sums: essentially 5*A081567.

Links

Table of n, a(n) for n=0..54.

Example

First eight rows:
    1,    -1;
    2,    -2,    1;
    5,    -6,    3,    -1;
   13,   -20,   12,    -4,   1;
   34,   -65,   50,   -20,   5,   -1;
   89,  -204,  195,  -100,  30,   -6,   1;
  233,  -623,  714,  -455, 175,  -42,   7, -1;
  610, -1864, 2492, -1904, 910, -280,  56, -8,  1;
First eight polynomials:
1 - x
2 - 2 x + x^2
5 - 6 x + 3 x^2 - x^3
13 - 20 x + 12 x^2 - 4 x^3 + x^4
34 - 65 x + 50 x^2 - 20 x^3 + 5 x^4 - x^5
89 - 204 x + 195 x^2 - 100 x^3 + 30 x^4 - 6 x^5 + x^6
233 - 623 x + 714 x^2 - 455 x^3 + 175 x^4 - 42 x^5 + 7 x^6 - x^7
610 - 1864 x + 2492 x^2 - 1904 x^3 + 910 x^4 - 280 x^5 + 56 x^6 - 8 x^7 + x^8

Mathematica

g[x_, n_] := Numerator[ Factor[D[(1 - x)/(x^2 - 3 x + 1), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x]
Table[h[n], {n, 0, 10}]
Column[%] (* A328646 array *)

Crossrefs

Cf. A328647, A001519, A000045, A081567.

Sequence in context: A010094 A019710 A118806 * A171670 A124644 A259691

Adjacent sequences: A328643 A328644 A328645 * A328647 A328648 A328649

Keyword

tabf,sign

Author

Clark Kimberling, Nov 01 2019

Status

approved