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 A129462 Coefficients of the v=2 member of a family of certain orthogonal polynomials. 7
 1, -1, 1, 0, -2, 1, 0, -6, 1, 1, 0, -48, -4, 12, 1, 0, -720, -204, 208, 35, 1, 0, -17280, -7776, 5208, 1348, 74, 1, 0, -604800, -358560, 179688, 64580, 5138, 133, 1, 0, -29030400, -20839680, 8175744, 3888528, 400384, 14952, 216, 1, 0, -1828915200, -1516112640, 472666752, 291010032 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS For v>=1 the orthogonal polynomials p(n,v,x) have v integer zeros k*(k-1), k=1..v, for every n>=v. These zeros are from 2*A000217. Coefficients of p(n,v=2,x) (in the quoted Bruschi et al. paper p^{(\nu)}_n(x) of eqs. (4) and (8a),(8b)) in increasing powers of x. The v-family p(n,v,x) consists of characteristic polynomials of the tridiagonal M x M matrix V=V(M,v) with entries V_{m,n} given by v*(v-1)-(m-1)^2 - (v-m)^2 if n=m, m=1,...,M;(m-1)^2 if n=m-1, m=2,...,M; (v-m)^2 if n=m+1, m=1..M-1 and 0 else. p(n,v,x):=det(x*I_n-V(n,v) with the n dimensional unit matrix I_n. The column sequences give A019590, A129464, A129465, A129466 for m=0,1,2,3. p(n,v=2,x) has, for every n>=2, simple zeros for integers x=0 and x=2. p(2,2,x) has therefore only integer zeros 0 and 2. det(V(n,2))=0 for every n>=2. 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. W. Lang, First ten rows and more. FORMULA a(n,m)=[x^m]p(n,1,x), n>=0, with the three term recurrence for orthogonal polynomial systems of the form p(n,v,x) = (x+2*(n-1)^2-2*(v-1)*(n-1)-v+1)*p(n-1,v,x) -(n-1)^2*(n-1-v)^2*p(n-2,v,x), n>=1; p(-1,v,x)=0 and p(0,v,x)=1. Put v=2 here. Recurrence: a(n,m) = a(n-1,m-1)+(2*(n-1)^2-2*(v-1)*(n-1)-v+1)*a(n-1,m) -((n-1)^2*(n-1-v)^2)*a(n-2, m); a(n,m)=0 if n

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Last modified February 20 22:47 EST 2020. Contains 332086 sequences. (Running on oeis4.)