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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)).
5

%I #6 Nov 06 2019 19:16:09

%S 1,-1,2,-2,1,5,-6,3,-1,13,-20,12,-4,1,34,-65,50,-20,5,-1,89,-204,195,

%T -100,30,-6,1,233,-623,714,-455,175,-42,7,-1,610,-1864,2492,-1904,910,

%U -280,56,-8,1,1597,-5490,8388,-7476,4284,-1638,420,-72,9,-1,4181

%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 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 First eight rows:

%e 1, -1;

%e 2, -2, 1;

%e 5, -6, 3, -1;

%e 13, -20, 12, -4, 1;

%e 34, -65, 50, -20, 5, -1;

%e 89, -204, 195, -100, 30, -6, 1;

%e 233, -623, 714, -455, 175, -42, 7, -1;

%e 610, -1864, 2492, -1904, 910, -280, 56, -8, 1;

%e First eight polynomials:

%e 1 - x

%e 2 - 2 x + x^2

%e 5 - 6 x + 3 x^2 - x^3

%e 13 - 20 x + 12 x^2 - 4 x^3 + x^4

%e 34 - 65 x + 50 x^2 - 20 x^3 + 5 x^4 - x^5

%e 89 - 204 x + 195 x^2 - 100 x^3 + 30 x^4 - 6 x^5 + x^6

%e 233 - 623 x + 714 x^2 - 455 x^3 + 175 x^4 - 42 x^5 + 7 x^6 - x^7

%e 610 - 1864 x + 2492 x^2 - 1904 x^3 + 910 x^4 - 280 x^5 + 56 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[%] (* A328646 array *)

%Y Cf. A328647, A001519, A000045, A081567.

%K tabf,sign

%O 0,3

%A _Clark Kimberling_, Nov 01 2019