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A328610 Irregular triangular array read by rows: the rows show the coefficients of the first of two factors of even-degree polynomials described in Comments.

%I A328610 #12 Oct 31 2019 21:42:22
%S A328610 -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 A328610 63,70,35,21,0,1,13,64,140,168,140,56,28,0,1,21,117,288,420,378,252,
%U A328610 84,36,0,1,34,210,585,960,1050,756,420,120,45,0,1,55,374
%N A328610 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 A328610 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 A328610 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 A328610 Clark Kimberling, <a href="/A328610/b328610.txt">Table of n, a(n) for n = 1..1325</a>
%e A328610 First nine rows:
%e A328610 .
%e A328610   -2,   1;  (coefficients of -2 + x)
%e A328610    1,   0,   1;  (coefficients of 1 + x^2)
%e A328610    1,   3,   0,   1;
%e A328610    2,   4,   6,   0,   1;
%e A328610    3,  10,  10,  10,   0,   1;
%e A328610    5,  18,  30,  20,  15,   0,   1;
%e A328610    8,  35,  63,  70,  35,  21,   0,   1;
%e A328610   13,  64, 140, 168, 140,  56,  28,   0,   1;
%e A328610   21, 117, 288, 420, 378, 252,  84,  36,   0,   1;
%t A328610 g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]];
%t A328610 f = Table[FactorList[g[x, n]/n!], {n, 1, 60, 2}]; (* polynomials *)
%t A328610 r[n_] := Rest[f[[n]]];
%t A328610 Column[Table[First[CoefficientList[r[n][[1]], x]], {n, 1, 16}]]  (* A328610 *)
%t A328610 Column[Table[-First[CoefficientList[r[n][[2]], x]], {n, 1, 16}]] (* A328611 *)
%Y A328610 Cf. A000045, A326925, A328611.
%K A328610 sign,tabf
%O A328610 1,1
%A A328610 _Clark Kimberling_, Oct 24 2019