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 A130182 Coefficients of the v=1 member of a family of certain orthogonal polynomials. 6
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

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Last modified May 20 01:04 EDT 2022. Contains 353847 sequences. (Running on oeis4.)