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%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
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