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A328647 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)).

%I A328647 #9 Jun 04 2022 22:03:41
%S A328647 1,1,4,-2,-1,11,-12,3,1,29,-44,24,-4,-1,76,-145,110,-40,5,1,199,-456,
%T A328647 435,-220,60,-6,-1,521,-1393,1596,-1015,385,-84,7,1,1364,-4168,5572,
%U A328647 -4256,2030,-616,112,-8,-1,3571,-12276,18756,-16716,9576,-3654,924,-144
%N A328647 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 A328647 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 A328647 First eight rows:
%e A328647      1,     1;
%e A328647      4,    -2,   -1;
%e A328647     11,   -12,    3,     1;
%e A328647     29,   -44,   24,    -4,   -1;
%e A328647     76,  -145,  110,   -40,    5,    1;
%e A328647    199,  -456,  435,  -220,   60,   -6,  -1;
%e A328647    521, -1393, 1596, -1015,  385,  -84,   7,  1;
%e A328647   1364, -4168, 5572, -4256, 2030, -616, 112, -8, -1;
%e A328647 First eight polynomials:
%e A328647 1 + x
%e A328647 4 - 2 x - x^2
%e A328647 11 - 12 x + 3 x^2 + x^3
%e A328647 29 - 44 x + 24 x^2 - 4 x^3 - x^4
%e A328647 76 - 145 x + 110 x^2 - 40 x^3 + 5 x^4 + x^5
%e A328647 199 - 456 x + 435 x^2 - 220 x^3 + 60 x^4 - 6 x^5 - x^6
%e A328647 521 - 1393 x + 1596 x^2 - 1015 x^3 + 385 x^4 - 84 x^5 + 7 x^6 + x^7
%e A328647 1364 - 4168 x + 5572 x^2 - 4256 x^3 + 2030 x^4 - 616 x^5 + 112 x^6 - 8 x^7 - x^8
%t A328647 g[x_, n_] := Numerator[ Factor[D[(1 + x)/(x^2 - 3 x + 1), {x, n}]]]
%t A328647 Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
%t A328647 h[n_] := CoefficientList[g[x, n]/n!, x]
%t A328647 Table[h[n], {n, 0, 10}]
%t A328647 Column[%]  (* A328647 array *)
%Y A328647 Cf. A328646, A002879, A000032.
%K A328647 tabf,sign
%O A328647 0,3
%A A328647 _Clark Kimberling_, Nov 01 2019