A328650 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-x-2x^2).

-1, 1, 4, -3, -6, -12, 5, 24, 24, 32, -11, -50, -120, -80, -80, 21, 132, 300, 480, 240, 192, -43, -294, -924, -1400, -1680, -672, -448, 85, 688, 2352, 4928, 5600, 5376, 1792, 1024, -171, -1530, -6192, -14112, -22176, -20160, -16128, -4608, -2304, 341, 3420

Offset: 0

Author

Clark Kimberling, Nov 01 2019

Keywords

• tabl
• sign

Comments

It appears that the number of nonconstant polynomial divisors of the n-th polynomial is given by A032741.

Examples

			First eight rows:
    -1;
     1,    4;
     3,   -6,  -12;
     5,   24,   24,    32;
   -11,  -50, -120,   -80,   -80;
    21,  132,  300,   480,   240,  192;
   -43, -294, -924, -1400, -1680, -672, -448;
    85,  688, 2352,  4928,  5600, 5376, 1792, 1024;
First eight polynomials:
-1
1 + 4 x
-3 (1 + 2 x + 4 x^2)
(1 + 4 x) (5 + 4 x + 8 x^2)
-11 - 50 x - 120 x^2 - 80 x^3 - 80 x^4
3 (1 + 4 x) (1 + 2 x + 4 x^2) (7 + 2 x + 4 x^2)
-43 - 294 x - 924 x^2 - 1400 x^3 - 1680 x^4 - 672 x^5 - 448 x^6
(1 + 4 x) (5 + 4 x + 8 x^2) (17 + 56 x + 120 x^2 + 32 x^3 + 32 x^4)
		

Crossrefs

Cf. A000032, A328646.

Programs

Mathematica

g[x_, n_] := Numerator[ Factor[D[1/(1 - x - 2 x^2), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]]  (* A328650 polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x];
Table[h[n], {n, 0, 10}] (* A328650 sequence *)
Column[%] (* A328650 array *)