A328646 - OEIS

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

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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