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 A129065 Coefficients of the v=1 member of a family of certain orthogonal polynomials. 10
 1, 0, 1, 0, 2, 1, 0, 12, 10, 1, 0, 144, 156, 28, 1, 0, 2880, 3696, 908, 60, 1, 0, 86400, 125280, 37896, 3508, 110, 1, 0, 3628800, 5780160, 2036592, 236472, 10528, 182, 1, 0, 203212800, 349090560, 138517632, 19022736, 1074176, 26600, 280, 1, 0 (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. Coefficients of p(n,v=1,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. p(n,v=1,x) has, for every n>=1 a zero for x=0, i.e. det(V(n,1))=0 for every n>=1. This is obvious. The column sequences give A000007, A010790, A129460, A129461 for m=0,1,2,3. 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. 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=1 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 December 7 18:19 EST 2021. Contains 349585 sequences. (Running on oeis4.)