login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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<x<1

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.

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.

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.

EXAMPLE

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;

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-Fran├žois Alcover, May 24 2016 *)

PROG

(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 */

CROSSREFS

Sequence in context: A152027 A077308 A075001 * A093803 A285175 A016599

Adjacent sequences:  A252837 A252838 A252839 * A252841 A252842 A252843

KEYWORD

sign,easy,tabl,changed

AUTHOR

Russell Jay Hendel, Dec 22 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 25 08:08 EST 2022. Contains 350565 sequences. (Running on oeis4.)