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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 (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=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 Adjacent sequences:  A252837 A252838 A252839 * A252841 A252842 A252843 KEYWORD sign,easy,tabl,changed AUTHOR Russell Jay Hendel, Dec 22 2014 STATUS approved

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