A328611 - OEIS

%I #11 Oct 31 2019 21:42:33

%S 0,-1,1,4,-1,3,3,6,-1,4,12,6,8,-1,7,20,30,10,10,-1,11,42,60,60,15,12,

%T -1,18,77,147,140,105,21,14,-1,29,144,308,392,280,168,28,16,-1,47,261,

%U 648,924,882,504,252,36,18,-1,76,470,1305,2160,2310,1764,840

%N Irregular triangular array read by rows: row n gives the coefficients of the second 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 Lucas numbers, L(k), for k >= 1; see A000032.  It appears that after the first row, the row sums are L(2k+1), for k >= 1.

%H Clark Kimberling, <a href="/A328611/b328611.txt">Table of n, a(n) for n = 1..1325</a>

%e First nine rows:

%e .

%e    0,  -1; (coefficients of -x)

%e    1,   4,  -1; (coefficients of 1 + 4*x - x^2)

%e    3,   3,   6,  -1;

%e    4,  12,   6,   8,  -1;

%e    7,  20,  30,  10,  10,  -1;

%e   11,  42,  60,  60,  15,  12,  -1;

%e   18,  77, 147, 140, 105,  21,  14, -1;

%e   29, 144, 308, 392, 280, 168,  28, 16, -1;

%e   47, 261, 648, 924, 882, 504, 252, 36, 18, -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. A000032, A326925, A328610.

%K tabf,sign

%O 1,4

%A _Clark Kimberling_, Oct 24 2019