A328647 - OEIS

%I #9 Jun 04 2022 22:03:41

%S 1,1,4,-2,-1,11,-12,3,1,29,-44,24,-4,-1,76,-145,110,-40,5,1,199,-456,

%T 435,-220,60,-6,-1,521,-1393,1596,-1015,385,-84,7,1,1364,-4168,5572,

%U -4256,2030,-616,112,-8,-1,3571,-12276,18756,-16716,9576,-3654,924,-144

%N 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)).

%C The first 201 polynomials are irreducible. Column 1 of the array: A002879 (odd-indexed Lucas numbers).  Row sums: A000032 (Lucas numbers). Alternating row sums: essentially 5*A030191.

%e First eight rows:

%e      1,     1;

%e      4,    -2,   -1;

%e     11,   -12,    3,     1;

%e     29,   -44,   24,    -4,   -1;

%e     76,  -145,  110,   -40,    5,    1;

%e    199,  -456,  435,  -220,   60,   -6,  -1;

%e    521, -1393, 1596, -1015,  385,  -84,   7,  1;

%e   1364, -4168, 5572, -4256, 2030, -616, 112, -8, -1;

%e First eight polynomials:

%e 1 + x

%e 4 - 2 x - x^2

%e 11 - 12 x + 3 x^2 + x^3

%e 29 - 44 x + 24 x^2 - 4 x^3 - x^4

%e 76 - 145 x + 110 x^2 - 40 x^3 + 5 x^4 + x^5

%e 199 - 456 x + 435 x^2 - 220 x^3 + 60 x^4 - 6 x^5 - x^6

%e 521 - 1393 x + 1596 x^2 - 1015 x^3 + 385 x^4 - 84 x^5 + 7 x^6 + x^7

%e 1364 - 4168 x + 5572 x^2 - 4256 x^3 + 2030 x^4 - 616 x^5 + 112 x^6 - 8 x^7 - x^8

%t g[x_, n_] := Numerator[ Factor[D[(1 + x)/(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}]

%t Column[%]  (* A328647 array *)

%Y Cf. A328646, A002879, A000032.

%K tabf,sign

%O 0,3

%A _Clark Kimberling_, Nov 01 2019