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A328646 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 A328646 #6 Nov 06 2019 19:16:09
%S A328646 1,-1,2,-2,1,5,-6,3,-1,13,-20,12,-4,1,34,-65,50,-20,5,-1,89,-204,195,
%T A328646 -100,30,-6,1,233,-623,714,-455,175,-42,7,-1,610,-1864,2492,-1904,910,
%U A328646 -280,56,-8,1,1597,-5490,8388,-7476,4284,-1638,420,-72,9,-1,4181
%N A328646 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 A328646 The first 501 polynomials are irreducible. Column 1 of the array: A001519 (odd-indexed Fibonacci numbers). Row sums: A000045 (Fibonacci numbers). Alternating row sums: essentially 5*A081567.
%e A328646 First eight rows:
%e A328646     1,    -1;
%e A328646     2,    -2,    1;
%e A328646     5,    -6,    3,    -1;
%e A328646    13,   -20,   12,    -4,   1;
%e A328646    34,   -65,   50,   -20,   5,   -1;
%e A328646    89,  -204,  195,  -100,  30,   -6,   1;
%e A328646   233,  -623,  714,  -455, 175,  -42,   7, -1;
%e A328646   610, -1864, 2492, -1904, 910, -280,  56, -8,  1;
%e A328646 First eight polynomials:
%e A328646 1 - x
%e A328646 2 - 2 x + x^2
%e A328646 5 - 6 x + 3 x^2 - x^3
%e A328646 13 - 20 x + 12 x^2 - 4 x^3 + x^4
%e A328646 34 - 65 x + 50 x^2 - 20 x^3 + 5 x^4 - x^5
%e A328646 89 - 204 x + 195 x^2 - 100 x^3 + 30 x^4 - 6 x^5 + x^6
%e A328646 233 - 623 x + 714 x^2 - 455 x^3 + 175 x^4 - 42 x^5 + 7 x^6 - x^7
%e A328646 610 - 1864 x + 2492 x^2 - 1904 x^3 + 910 x^4 - 280 x^5 + 56 x^6 - 8 x^7 + x^8
%t A328646 g[x_, n_] := Numerator[ Factor[D[(1 - x)/(x^2 - 3 x + 1), {x, n}]]]
%t A328646 Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
%t A328646 h[n_] := CoefficientList[g[x, n]/n!, x]
%t A328646 Table[h[n], {n, 0, 10}]
%t A328646 Column[%] (* A328646 array *)
%Y A328646 Cf. A328647, A001519, A000045, A081567.
%K A328646 tabf,sign
%O A328646 0,3
%A A328646 _Clark Kimberling_, Nov 01 2019