A328645 - OEIS

%I #6 Nov 06 2019 19:15:59

%S 1,3,-2,8,-9,3,21,-32,18,-4,55,-105,80,-30,5,144,-330,315,-160,45,-6,

%T 377,-1008,1155,-735,280,-63,7,987,-3016,4032,-3080,1470,-448,84,-8,

%U 2584,-8883,13572,-12096,6930,-2646,672,-108,9,6765,-25840,44415,-45240

%N 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)).

%C 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.

%e First eight rows:

%e     1;

%e     3,    -2;

%e     8,    -9,    3;

%e    21,   -32,   18,    -4;

%e    55,  -105,   80,   -30,    5;

%e   144,  -330,  315,  -160,   45,   -6;

%e   377, -1008, 1155,  -735,  280,  -63,  7;

%e   987, -3016, 4032, -3080, 1470, -448, 84, -8;

%e First eight polynomials:

%e 1

%e 3 - 2 x

%e 8 - 9 x + 3 x^2

%e 21 - 32 x + 18 x^2 - 4 x^3

%e       = (3 - 2 x) (7 - 6 x + 2 x^2)

%e 55 - 105 x + 80 x^2 - 30 x^3 + 5 x^4

%e 144 - 330 x + 315 x^2 - 160 x^3 + 45 x^4 - 6 x^5

%e       = (3 - 2 x) (6 - 3 x + x^2) (8 - 9 x + 3 x^2)

%e 377 - 1008 x + 1155 x^2 - 735 x^3 + 280 x^4 - 63 x^5 + 7 x^6

%e 987 - 3016 x + 4032 x^2 - 3080 x^3 + 1470 x^4 - 448 x^5 + 84 x^6 - 8 x^7

%e       = (3 - 2 x) (7 - 6 x + 2 x^2) (47 - 72 x + 42 x^2 - 12 x^3 + 2 x^4)

%t g[x_, n_] := Numerator[ Factor[D[1/(x^2 - 3 x + 1), {x, n}]]]

%t Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)

%t h[n_] := CoefficientList[g[x, n]/n!, x]

%t Table[h[n], {n, 0, 10}] (* A328645 array *)

%Y Cf. A326933.

%K tabl,sign

%O 0,2

%A _Clark Kimberling_, Nov 01 2019