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%I #12 Oct 31 2019 21:42:22
%S -2,1,1,0,1,1,3,0,1,2,4,6,0,1,3,10,10,10,0,1,5,18,30,20,15,0,1,8,35,
%T 63,70,35,21,0,1,13,64,140,168,140,56,28,0,1,21,117,288,420,378,252,
%U 84,36,0,1,34,210,585,960,1050,756,420,120,45,0,1,55,374
%N Irregular triangular array read by rows: the rows show the coefficients of the first of two factors of even-degree polynomials described in Comments.
%C Let p(n) denote the polynomial (1/n!)*(numerator of n-th derivative of (1-x)/(1-x-x^2)). It is conjectured in A326925 that if n = 2k, then p(n) = f(k)*g(k), where f(k) and g(k) are polynomials of degree k. Row k of the present array shows the coefficients of f(k).
%C It appears that, after the first term, column 1 consists of the Fibonacci numbers, F(k), for k >= 1; see A000045. It appears that after the first row, the row sums are F(2k+1), and the alternating row sums are (-1)^k F(k).
%H Clark Kimberling, <a href="/A328610/b328610.txt">Table of n, a(n) for n = 1..1325</a>
%e First nine rows:
%e .
%e -2, 1; (coefficients of -2 + x)
%e 1, 0, 1; (coefficients of 1 + x^2)
%e 1, 3, 0, 1;
%e 2, 4, 6, 0, 1;
%e 3, 10, 10, 10, 0, 1;
%e 5, 18, 30, 20, 15, 0, 1;
%e 8, 35, 63, 70, 35, 21, 0, 1;
%e 13, 64, 140, 168, 140, 56, 28, 0, 1;
%e 21, 117, 288, 420, 378, 252, 84, 36, 0, 1;
%t g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]];
%t f = Table[FactorList[g[x, n]/n!], {n, 1, 60, 2}]; (* polynomials *)
%t r[n_] := Rest[f[[n]]];
%t Column[Table[First[CoefficientList[r[n][[1]], x]], {n, 1, 16}]] (* A328610 *)
%t Column[Table[-First[CoefficientList[r[n][[2]], x]], {n, 1, 16}]] (* A328611 *)
%Y Cf. A000045, A326925, A328611.
%K sign,tabf
%O 1,1
%A _Clark Kimberling_, Oct 24 2019
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