A252840 Coefficients of G_i(x) with G_0 = 1, G_1 = 1+x, G_n = (1-2*x)*G_{n-1}+(x-x^2)*G_{n-2}.

1, 1, 1, 1, 0, -3, 1, -1, -3, 5, 1, -2, -2, 8, -7, 1, -3, 0, 10, -15, 9, 1, -4, 3, 10, -25, 24, -11, 1, -5, 7, 7, -35, 49, -35, 13, 1, -6, 12, 0, -42, 84, -84, 48, -15, 1, -7, 18, -12, -42, 126, -168, 132, -63, 17

Offset: 0

Author

Russell Jay Hendel, Dec 22 2014

Keywords

• sign
• easy
• tabl

Comments

There are 3 equivalent ways to define this sequence:

#1) Recursively defined polynomial sequence:

G_0(x)=1, G_1(x)=1+x; G_n=(1-2x)G_{n-1}+(x-x^2)G_{n-2}. One can then form a triangle whose n-th row lists the coefficients of G_n(x) starting from the constant term. The list of these rows forms the sequence. It follows that an explicit formula for G_n is G_n(x)=(1+2x)(1-x)^n-2x(-x)^n.

#2) Recursively defined triangle:

Define an array g^(i)n as follows: i) Leftmost column: g^(0)_n=1, n >=0; ii) Define the diagonal sequence D_n = g^(n)_n by D_0=0, D_1=1 and D_n=-2D{n-1}- D_{n-2}; iii) Boundary conditions: g^(i)n=0 if i<0, n<0 or n>i; iv) g^(i)_n - g^(i){n-1}= -g^(i-1)_{n-1} for n>=2, 1 <=i <=n-1. It follows that g^(i)_n = -(-1)^j(2 C(n,j-1)-C(n,j)) with C(n,j) the binomial coefficient.

#3) Polynomial columns:

g^(0)n =1 (n >=0); g^(i)_n= (-1)^i 1/i! (n-(3i-1))(n){i-1} for i>=1 and n>=i (g^(i)n is 0 elsewhere). Here (n)_m is the falling factorial with (n)_0=1 and (n){m+1}=(n)_m (n-m).

The polynomials G_n = (1+2x)(1-x)^n-2x(-x)^n have interesting limit properties. (All limits are for fixed x as n goes to infinity.)

1) lim G_n(-1) = lim -(2^n-2) = minus infinity

2) lim G_n(x) = minus infinity, for -1 <= x < -1/2

3) lim G_n(-1/2) = lim (1/2)^n =0

4) lim G_n(x) = positive infinity, for -1/2 < x <0

5) lim G_n(0) = 1 (in fact G_n(0)=1 for all n)

6) lim G_n(x) = 0, for 0

7) lim G_n(1) = lim -(-1)^n 2, diverges by oscillation

There are other interesting limits. For example, although for all n, G_n(0)=1, nevertheless lim G_n(1/n) = 1/e.

Examples

			G_0(x)=1, G_1(x)=1+x; G_2(x)=(1-2x)G_1(x)+(x-x^2)G_0(x)=1-3x^2. Hence the 0th row is 1; the 1st row is 1,1; the 2nd row is 1,0,-3.
Similarly we may give the triangle by columns: For example, the 2nd column is described by the polynomial (-1)^2 1/2! (n-(3*2-1))(n)_1 = -1/2(n-5)n which for n>=2 generates the values -3,-3,-2,0,3,7,12 which are the values of the 2nd column of the triangle starting from the 2nd row.
The first few rows of the triangle are
  1;
  1,   1;
  1,   0,   -3;
  1,  -1,   -3,  5;
  1,  -2,   -2,  8,  -7;
  1,  -3,    0, 10,  -15,  9;
		

References

• Russell Jay Hendel, "Polynomial Convergence of Recursively Defined Polynomials", West Coast Number Theory Conference, Pacific Grove, 2014.

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Russell Jay Hendel, Coefficient Convergence of Recursively Defined Polynomials, Fibonacci Quart. 53 (2015), no. 3, 247-252.
• Clark Kimberling, Limits of Polynomial Sequences, Fibonacci Quarterly, 50(4), 2012, pp. 294-297.

Programs

Maple

g:= proc(n) option remember; `if`(n<2, 1+n*x,
      expand((1-2*x)*g(n-1)+(x-x^2)*g(n-2)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(g(n)):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 22 2014

Mathematica

row[n_] := CoefficientList[(1+2x)(1-x)^n-2x(-x)^n, x]; Array[row, 10, 0] // Flatten (* Jean-François Alcover, May 24 2016 *)

PARI

/* using FORMULA #2 */
/* Function Triangle produces first n rows of coefficients of the recursively defined polynomials */
Triangle(n)={
m=matrix(n,n,i,j,0);
/* Leftmost column is 1*/
for(i=1,n,m[i,1]=1);
/* Rightmost diagonal are odds with alternating sign */
for(i=1,n,m[i,i]=(-1)^i *(2*i-3));
/* Body of triangle based on recursive formula */
for(i=3,n,for(j=2,i-1,m[i,j]=m[i-1,j]-m[i-1,j-1]));
m
}
/* By calling Triangle(10) one obtains the first 10 rows of the triangle corresponding to the entry in the DATA step */

Formula

Method #1) G_0(x)=1; G_1(x)=1+x; G_n(x)=(1-2x)G_{n-1}(x) + (x-x^2) G_{n-2}(x). The n-th row of the triangle lists the coefficients of G_n(x)=(1+2x)(1-x)^n-2x(-x)^n.

Method #2) g^(0)n=1; D=g^(n)_n with D_0=1=D_1 and D_n=-2D{n-1}- D_{n-2}; g^(i)n =g^(i){n-1} - g^(i-1)_{n-1}, n >=2 and 1 <= i <= n-1. Hence g^(i)_n = -(-1)^i (2C(n,i-1)-C(n,i))

Method #3) g^(0)n=1; g^(i)_n= (-1)^i 1/i! (n-(3i-1))(n){i-1}, i>=1, n>=i, with (n)_m the falling factorial. For each i, the g^(i)_n are a polynomial of degree i describing the triangle entry in row n and column i with n>=i.

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