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A326925 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)/(1-x-x^2)).

%I A326925 #13 Oct 31 2019 21:41:49
%S A326925 1,-1,0,2,-1,1,0,3,-1,1,4,0,4,-1,2,5,10,0,5,-1,3,12,15,20,0,6,-1,5,21,
%T A326925 42,35,35,0,7,-1,8,40,84,112,70,56,0,8,-1,13,72,180,252,252,126,84,0,
%U A326925 9,-1,21,130,360,600,630,504,210,120,0,10,-1,34,231,715
%N A326925 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)/(1-x-x^2)).
%C A326925 Column 1: Fibonacci numbers, F(m), for m >= -1, as in A000045. For n >= 0, the n-th row sum = F(2n), as in A001906.
%C A326925 Conjecture: The odd degree polynomials are irreducible; the even degree (= 2k) polynomials have exactly two irreducible factors, each of degree k.
%H A326925 Clark Kimberling, <a href="/A326925/b326925.txt">Table of n, a(n) for n = 1..10010</a>
%F A326925 G.f. as array: ((y^2 + y - 1)*x - y + 1)/(1 + (y^2 + y - 1)*x^2 + (-2*y - 1)*x). - _Robert Israel_, Oct 31 2019
%e A326925 First 7 rows:
%e A326925 1    -1
%e A326925 0     2   -1
%e A326925 1     0    3   -1
%e A326925 1     4    0    4   -1
%e A326925 2     5    0   10    5   -1
%e A326925 3    12   15   20    0    6   -1
%e A326925 5    21   42   35   35    0    7   -1
%e A326925 First 7 polynomials:
%e A326925 1 - x
%e A326925 2 x - x^2
%e A326925 1 + 3 x^2 - x^3
%e A326925 1 + 4 x + 4 x^3 - x^4
%e A326925 2 + 5 x + 10 x^2 + 5 x^4 - x^5
%e A326925 3 + 12 x + 15 x^2 + 20 x^3 + 6 x^5 - x^6
%e A326925 5 + 21 x + 42 x^2 + 35 x^3 + 35 x^4 + 7 x^6 - x^7
%e A326925 Factorizations of even-degree polynomials:
%e A326925 degree 2:  (2 - x)*x
%e A326925 degree 4:  (1 + x^2)*(1 + 4x - x^2)
%e A326925 degree 6:  (1 + 3x + x^3)*(3 + 3x + 6x^2 - x^3)
%e A326925 degree 8:  (2 + 4x + 6x^2 + x^4)*(4 + 12 x + 6x^2 + 8x^3 - x^4)
%e A326925 degree 10: (3 + 10 x + 10 x^2 + 10 x^3 + x^5)*(7 + 20 x + 30 x^2 + 10 x^3 + 10 x^4 - x^5)
%e A326925 It appears that the constant terms of the factors are Fibonacci numbers (A000045) and Lucas numbers (A000032).
%t A326925 g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]]
%t A326925 Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
%t A326925 h[n_] := CoefficientList[g[x, n]/n!, x] (* A326925 *)
%t A326925 Table[h[n], {n, 0, 10}]
%t A326925 Column[%]
%Y A326925 Cf. A000045, A001906, A094440, A094441, A328610, A328611.
%K A326925 tabf,sign
%O A326925 1,4
%A A326925 _Clark Kimberling_, Oct 22 2019