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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

Table of n, a(n) for n=0..45.

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<m, a(-1,m):=0, a(0,0)=1, a(n,-1)=0. Put v=1 here.

EXAMPLE

Triangle begins:

[1];

[0,1];

[0,2,1];

[0,12,10,1];

[0,144,156,28,1];

[0,2880,3696,908,60,1];

...

n=5,[0,2880,3696,908,60,1] stands for the polynomial x*(2880+3696*x+908*x^2+60*x^3+1*x^4) with one zero 0 and some other four zeros.

Tridiagonal matrix V(5,1)=[[0,0,0,0,0],[1,-2,1,0,0],[0,4,-8,4,0],[0,0,9,-18,9],[0,0,0,16,-32]].

MATHEMATICA

nmax = 9; a[n_, m_] := a[n, m] = (-(n-2)^2)*(n-1)^2*a[n-2, m] + a[n-1, m-1] + 2*(n-1)^2*a[n-1, m]; a[n_, m_] /; n < m = 0; a[-1, _] = 0; a[0, 0] = 1; a[_, -1] = 0; Flatten[ Table[ a[n, m], {n, 0, nmax}, {m, 0, n}]] (* Jean-François Alcover, Sep 26 2011, after recurrence *)

CROSSREFS

Row sums give A129458. Cf. A129462 (v=2 triangle).

Sequence in context: A268435 A039910 A129467 * A202700 A024026 A269946

Adjacent sequences:  A129062 A129063 A129064 * A129066 A129067 A129068

KEYWORD

nonn,tabl,easy

AUTHOR

Wolfdieter Lang, May 04 2007

STATUS

approved

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Last modified December 7 18:19 EST 2021. Contains 349585 sequences. (Running on oeis4.)