A328645 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/(1-3x+x^2)).

1, 3, -2, 8, -9, 3, 21, -32, 18, -4, 55, -105, 80, -30, 5, 144, -330, 315, -160, 45, -6, 377, -1008, 1155, -735, 280, -63, 7, 987, -3016, 4032, -3080, 1470, -448, 84, -8, 2584, -8883, 13572, -12096, 6930, -2646, 672, -108, 9, 6765, -25840, 44415, -45240

Offset

0,2

Comments

It appears that (number of nonconstant polynomial divisors of the n-th degree polynomial) = A032741(n+1) = number of divisors d of n+1 that are < n+1, for n >= 0.

Links

Table of n, a(n) for n=0..48.

Example

First eight rows:
    1;
    3,    -2;
    8,    -9,    3;
   21,   -32,   18,    -4;
   55,  -105,   80,   -30,    5;
  144,  -330,  315,  -160,   45,   -6;
  377, -1008, 1155,  -735,  280,  -63,  7;
  987, -3016, 4032, -3080, 1470, -448, 84, -8;
First eight polynomials:
1
3 - 2 x
8 - 9 x + 3 x^2
21 - 32 x + 18 x^2 - 4 x^3
      = (3 - 2 x) (7 - 6 x + 2 x^2)
55 - 105 x + 80 x^2 - 30 x^3 + 5 x^4
144 - 330 x + 315 x^2 - 160 x^3 + 45 x^4 - 6 x^5
      = (3 - 2 x) (6 - 3 x + x^2) (8 - 9 x + 3 x^2)
377 - 1008 x + 1155 x^2 - 735 x^3 + 280 x^4 - 63 x^5 + 7 x^6
987 - 3016 x + 4032 x^2 - 3080 x^3 + 1470 x^4 - 448 x^5 + 84 x^6 - 8 x^7
      = (3 - 2 x) (7 - 6 x + 2 x^2) (47 - 72 x + 42 x^2 - 12 x^3 + 2 x^4)

Mathematica

g[x_, n_] := Numerator[ Factor[D[1/(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}] (* A328645 array *)

Crossrefs

Cf. A326933.

Sequence in context: A230432 A195305 A327575 * A021308 A274181 A195055

Adjacent sequences: A328642 A328643 A328644 * A328646 A328647 A328648

Keyword

tabl,sign

Author

Clark Kimberling, Nov 01 2019

Status

approved