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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
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
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-François 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
KEYWORD
sign,easy,tabl
AUTHOR
Russell Jay Hendel, Dec 22 2014
STATUS
approved