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
Examples
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).
Links
- Clark Kimberling, Table of n, a(n) for n = 1..10010
Programs
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Mathematica
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[%]
Formula
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
Comments