A328610 -id:A328610 - cp's OEIS frontend

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)).

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, 42, 35, 35, 0, 7, -1, 8, 40, 84, 112, 70, 56, 0, 8, -1, 13, 72, 180, 252, 252, 126, 84, 0, 9, -1, 21, 130, 360, 600, 630, 504, 210, 120, 0, 10, -1, 34, 231, 715

Offset: 1

Clark Kimberling, Oct 22 2019

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.

Conjecture: The odd degree polynomials are irreducible; the even degree (= 2k) polynomials have exactly two irreducible factors, each of degree k.

			First 7 rows:
1    -1
0     2   -1
1     0    3   -1
1     4    0    4   -1
2     5    0   10    5   -1
3    12   15   20    0    6   -1
5    21   42   35   35    0    7   -1
First 7 polynomials:
1 - x
2 x - x^2
1 + 3 x^2 - x^3
1 + 4 x + 4 x^3 - x^4
2 + 5 x + 10 x^2 + 5 x^4 - x^5
3 + 12 x + 15 x^2 + 20 x^3 + 6 x^5 - x^6
5 + 21 x + 42 x^2 + 35 x^3 + 35 x^4 + 7 x^6 - x^7
Factorizations of even-degree polynomials:
degree 2:  (2 - x)*x
degree 4:  (1 + x^2)*(1 + 4x - x^2)
degree 6:  (1 + 3x + x^3)*(3 + 3x + 6x^2 - x^3)
degree 8:  (2 + 4x + 6x^2 + x^4)*(4 + 12 x + 6x^2 + 8x^3 - x^4)
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)
It appears that the constant terms of the factors are Fibonacci numbers (A000045) and Lucas numbers (A000032).

Clark Kimberling, Table of n, a(n) for n = 1..10010

Cf. A000045, A001906, A094440, A094441, A328610, A328611.

g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x] (* A326925 *)
Table[h[n], {n, 0, 10}]
Column[%]

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

A328611 Irregular triangular array read by rows: row n gives the coefficients of the second of two factors of even-degree polynomials described in Comments.

Original entry on oeis.org

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, -1, 18, 77, 147, 140, 105, 21, 14, -1, 29, 144, 308, 392, 280, 168, 28, 16, -1, 47, 261, 648, 924, 882, 504, 252, 36, 18, -1, 76, 470, 1305, 2160, 2310, 1764, 840

Offset: 1

Clark Kimberling, Oct 24 2019

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).

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.

			First nine rows:
.
   0,  -1; (coefficients of -x)
   1,   4,  -1; (coefficients of 1 + 4*x - x^2)
   3,   3,   6,  -1;
   4,  12,   6,   8,  -1;
   7,  20,  30,  10,  10,  -1;
  11,  42,  60,  60,  15,  12,  -1;
  18,  77, 147, 140, 105,  21,  14, -1;
  29, 144, 308, 392, 280, 168,  28, 16, -1;
  47, 261, 648, 924, 882, 504, 252, 36, 18, -1;

Clark Kimberling, Table of n, a(n) for n = 1..1325

Cf. A000032, A326925, A328610.

g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]];
f = Table[FactorList[g[x, n]/n!], {n, 1, 60, 2}]; (* polynomials *)
r[n_] := Rest[f[[n]]];
Column[Table[First[CoefficientList[r[n][[1]], x]], {n, 1, 16}]]  (* A328610 *)
Column[Table[-First[CoefficientList[r[n][[2]], x]], {n, 1, 16}]] (* A328611 *)

cp's OEIS Frontend

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)).

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