%I A130182
%S A130182 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,
%T A130182 98,1,0,3628800,449280,788688,153760,8568,168,1,0,203212800,42405120,46032768,
%U A130182 11269008,811648,23016,264,1,0,14631321600,4187635200,3372731136
%V A130182 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,
%W A130182 98,1,0,-3628800,-449280,788688,153760,8568,168,1,0,-203212800,-42405120,
               46032768,
%X A130182 11269008,811648,23016,264,1,0,-14631321600,-4187635200,3372731136
%N A130182 Coefficients of the v=1 member of a family of certain orthogonal polynomials.
%C A130182 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.
%C A130182 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 A130182 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.
%C A130182 The column sequences give [1,-2,0,0,0,...], A010790(n-1)*(-1)^(n-1), 
               A130185, A130186 for m=0,1,2,3.
%C A130182 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.
%D A130182 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.
%H A130182 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A130182.text">
               First ten rows and more. </a>
%F A130182 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.
%F A130182 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.
%e A130182 [1]; [ -2,1]; [0,-2,1]; [0,-12,4,1]; [0,-144,28,20,1]; [0,-2880,216,508,
               50,1]; ...
%e A130182 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).
%e A130182 Row n=1:[ -2,1]. pt(1,1,x)=-2+x with integer zero x=2.
%Y A130182 Cf. A129065 (a v=1 member of a similar family).
%Y A130182 Row sums A130183, unsigned row sums A130184.
%Y A130182 Sequence in context: A138468 A029296 A096419 this_sequence A024361 A135486 
               A030187
%Y A130182 Adjacent sequences: A130179 A130180 A130181 this_sequence A130183 A130184 
               A130185
%K A130182 sign,tabl,easy
%O A130182 0,2
%A A130182 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jun 01 2007