A130559 - OEIS

A130559

Coefficients of the v=n member of a family of certain orthogonal polynomials with Diophantine properties.

1, -2, 1, 12, -8, 1, -144, 108, -20, 1, 2880, -2304, 508, -40, 1, -86400, 72000, -17544, 1708, -70, 1, 3628800, -3110400, 808848, -89280, 4648, -112, 1, -203212800, 177811200, -48405888, 5808528, -349568, 10920, -168, 1, 14631321600, -13005619200, 3663035136, -466619904

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

For v>=1 the orthogonal polynomials pt(n,v,x) have only integer zeros k*(k+1), k=1..n These integer zeros are from 2*A000217.

Coefficients of pt(n,v=n,x) (in the quoted Bruschi et al. paper {\tilde p}^{(\nu)}_n(x) of eqs. (20) and (24a),(24b)) in increasing powers of x.

The v-family pt(n,v,x) consists of characteristic polynomials of the tridiagonal M x M matrix Vt=Vt(M,v) with entries Vt_{m,n} given by 2*m*(v+1-m) if n=m, m=1,...,M; -m*(v+1-m) if n=m-1, m=2,...,M; -m*(v+1-m) if n=m+1, m=1..M-1 and 0 else. pt(n,v,x):=det(x*I_n-Vt(n,v) with the n dimensional unit matrix I_n.

pt(n,v=n,x) has, for every n>=1, the n integer zeros 2,6,12,...,n*(n+1). pt(2,2,x) has therefore only the integer zeros 2 and 6. 12= 2*6 = det(Vt(2,2))=16-4.

This triangle coincides with triangle A129467 without row n=0 and column m=0, taking as offset again [0,0].

Column sequences give for m=0..2: A010790(n-1)*(-1)^(n-1), A084915(n+1)*(-1)^n, A130033.

LINKS

Table of n, a(n) for n=0..39.

M. Bruschi, F. Calogero and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Physics A, 40(2007), pp. 3815-3829.

M. W. Coffey, M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.

W. Lang, First 10 rows and more.

FORMULA

a(n,m)=[x^m]pt(n,n,x), n>=0, with the three term recurrence for orthogonal polynomial systems of the form pt(n,v,x) = (x + 2*n*(n-1-v)*pt(n-1,v,x) -(n-1)*n*(n-1-v)*(n-2-v)*pt(n-2,v,x), n>=1; pt(-1,v,x)=0 and pt(0,v,x)=1. Start with v=n.

EXAMPLE

n=2: [12,-8,1 stands for pt(2,2,x)=12-8*x+x^2 = (x-2)*(x-6) with the integer zeros 2*1 and 2*3.

Triangle begins:

[1];

[-2,1];

[12,-8,1];

[-144,108,-20,1];

[2880,-2304,508,-40,1];

...

CROSSREFS

Row sums give A130031(n+1), n>=0. Unsigned row sums give A130032(n+1), n>=1.

Cf. A130182 (v=1 member).