A130182 Coefficients of the v=1 member of a family of certain orthogonal polynomials.

1, -2, 1, 0, -2, 1, 0, -12, 4, 1, 0, -144, 28, 20, 1, 0, -2880, 216, 508, 50, 1, 0, -86400, -2592, 17400, 2548, 98, 1, 0, -3628800, -449280, 788688, 153760, 8568, 168, 1, 0, -203212800, -42405120, 46032768, 11269008, 811648, 23016, 264, 1, 0, -14631321600, -4187635200, 3372731136

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..48.

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.

Wolfdieter Lang, First ten rows and more.

Formula

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.

Example

Triangle begins:
[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.

Crossrefs

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

Row sums A130183, unsigned row sums A130184.

Sequence in context: A029296 A096419 A283616 * A024361 A305614 A190676

Adjacent sequences: A130179 A130180 A130181 * A130183 A130184 A130185

Keyword

sign,tabl,easy

Author

Wolfdieter Lang, Jun 01 2007

Status

approved