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A130559 Coefficients of the v=n member of a family of certain orthogonal polynomials with Diophantine properties. 3


%S 1,-2,1,12,-8,1,-144,108,-20,1,2880,-2304,508,-40,1,-86400,72000,

%T -17544,1708,-70,1,3628800,-3110400,808848,-89280,4648,-112,1,

%U -203212800,177811200,-48405888,5808528,-349568,10920,-168,1,14631321600,-13005619200,3663035136,-466619904

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

%C 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.

%C 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.

%C 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.

%C 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.

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

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

%H M. Bruschi, F. Calogero and R. Droghei, <a href="http://dx.doi.org/10.1088/1751-8113/40/14/005">Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials</a>, J. Physics A, 40(2007), pp. 3815-3829.

%H M. W. Coffey, M. C. Lettington, <a href="http://arxiv.org/abs/1510.05402">On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat</a>, arXiv:1510.05402 [math.NT], 2015.

%H W. Lang, <a href="/A130559/a130559.txt">First 10 rows and more</a>.

%F 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.

%e 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.

%e Triangle begins:

%e [1];

%e [-2,1];

%e [12,-8,1];

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

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

%e ...

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

%Y Cf. A130182 (v=1 member).

%K sign,tabl,easy

%O 0,2

%A _Wolfdieter Lang_, Jul 13 2007

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Last modified August 7 09:16 EDT 2020. Contains 336274 sequences. (Running on oeis4.)