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A328649 Irregular triangular array read by rows: row n shows the coefficients of the following polynomial of degree n: (1/n!)*(numerator of n-th derivative of (x-2)/(1-x-x^2)).

%I A328649 #12 Dec 10 2023 17:45:43
%S A328649 2,-1,-1,-4,1,3,3,6,-1,-4,-12,-6,-8,1,7,20,30,10,10,-1,-11,-42,-60,
%T A328649 -60,-15,-12,1,18,77,147,140,105,21,14,-1,-29,-144,-308,-392,-280,
%U A328649 -168,-28,-16,1,47,261,648,924,882,504,252,36,18,-1,-76,-470,-1305
%N A328649 Irregular triangular array read by rows: row n shows the coefficients of the following polynomial of degree n: (1/n!)*(numerator of n-th derivative of (x-2)/(1-x-x^2)).
%C A328649 The first 201 polynomials are irreducible.  Column 1: signed Lucas numbers (A000032).
%e A328649 First eight rows:
%e A328649     2,   -1;
%e A328649    -1,   -4,    1;
%e A328649     3,    3,    6,   -1;
%e A328649    -4,  -12,   -6,   -8,    1;
%e A328649     7,   20,   30,   10,   10,   -1;
%e A328649   -11,  -42,  -60,  -60,  -15,  -12,   1;
%e A328649    18,   77,  147,  140,  105,   21,  14,  -1;
%e A328649   -29, -144, -308, -392, -280, -168, -28, -16, 1;
%e A328649 First eight polynomials:
%e A328649    2 - x
%e A328649   -1 - 4 x + x^2
%e A328649    3 + 3 x + 6 x^2 - x^3
%e A328649   -4 - 12 x - 6 x^2 - 8 x^3 + x^4
%e A328649    7 + 20 x + 30 x^2 + 10 x^3 + 10 x^4 - x^5
%e A328649   -11 - 42 x - 60 x^2 - 60 x^3 - 15 x^4 - 12 x^5 + x^6
%e A328649    18 + 77 x + 147 x^2 + 140 x^3 + 105 x^4 + 21 x^5 + 14 x^6 - x^7
%e A328649   -29 - 144 x - 308 x^2 - 392 x^3 - 280 x^4 - 168 x^5 - 28 x^6 - 16 x^7 + x^8
%t A328649 g[x_, n_] := Numerator[ Factor[D[(x - 2)/(1 - x - x^2), {x, n}]]]
%t A328649 Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]]  (* A328649 polynomials *)
%t A328649 h[n_] := CoefficientList[g[x, n]/n!, x];
%t A328649 Table[h[n], {n, 0, 10}] (* A328649 sequence *)
%t A328649 Column[%] (* A328649 array *)
%Y A328649 Cf. A000032, A328646.
%K A328649 tabf,sign
%O A328649 0,1
%A A328649 _Clark Kimberling_, Nov 01 2019