0,2

For v>=1 the orthogonal polynomials pt(n,v,x) have v integer zeros k*(k+1), k=1..v, for every n>=v and some other n-v zeros. The integer zeros are from 2*A000217.

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=1,x) has, for every n>=1, among its n zeros one for x=2. pt(1,1,x) has therefore only the integer zeros 2. det(Vt(1,1))=2.

The column sequences give [1,-2,0,0,0,...], A010790(n-1)*(-1)^(n-1), A130185, A130186 for m=0,1,2,3.

Coefficients of pt(n,v=1,x) (in the quoted Bruschi et al. paper {\tilde p}^{(\nu)}_n(x) of eqs. (20) and (24a),(24b)) in increasing powers of x.

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)3815-3829.

W. Lang, First ten rows and more.

a(n,m)=[x^m]pt(n,1,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. Put v=1 here.

Recurrence: a(n,m) = a(n-1,m-1)+2*n*(n-2)*a(n-1,m) - (n-1)*n*(n-2)*(n-3)*a(n-2,m); a(n,m)=0 if n<m, a(-1,m):=0, a(n,-1)=1. From the pt(n,1,x) recurrence.

[1]; [ -2,1]; [0,-2,1]; [0,-12,4,1]; [0,-144,28,20,1]; [0,-2880,216,508,50,1]; ...

Row n=5:[0,-2880,216,508,50,1]; pt(5,2,x)= x*(-2880+216*x+508*x^2+50*x^3+1*x^4)= x*(x-2)*(1440+612*x+52*x^2+x^3). pt(5,1,x) has the guaranteed integer zero x=2 (and also x=0 and some other three zeros).

Row n=1:[ -2,1]. pt(1,1,x)=-2+x with integer zero x=2.

Cf. A129065 (a v=1 member of a similar family).

Row sums A130183, unsigned row sums A130184.

Sequence in context: A138468 A029296 A096419 this_sequence A024361 A135486 A030187

Adjacent sequences: A130179 A130180 A130181 this_sequence A130183 A130184 A130185

sign,tabl,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jun 01 2007