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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

Table of n, a(n) for n=0..49.

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<m, a(-1,m):=0, a(0,0)=1, a(n,-1)=0. Put v=2 for this triangle.

EXAMPLE

Triangle begins:

[1];

[-1,1];

[0,-2,1];

[0,-6,1,1];

[0,-48,-4,12,1];

[0,-720,-204,208,35,1];

...

Row n=2: [0,-2,1]. p(2,2,x)=x*(x-2).

Row n=5: [0,-720,-204,208,35,1]. p(5,2,x)= x*(-720 -204*x+208*x^2+35*x^3+1*x^4)= x*(x-2)*(360+282*x+37*x^2+x^3).

MATHEMATICA

p[-1, _, _] = 0 ; p[0, _, _] = 1; p[n_, v_, x_] := 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]; a[n_, m_] := Coefficient[p[n, 2, x], x, m]; Table[a[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 30 2013 *)

CROSSREFS

Cf. A129065 (v=1 triangle). Row sums A129463.

Sequence in context: A196776 A157982 A119275 * A122930 A066387 A180663

Adjacent sequences:  A129459 A129460 A129461 * A129463 A129464 A129465

KEYWORD

sign,tabl,easy

AUTHOR

Wolfdieter Lang, May 04 2007

STATUS

approved

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Last modified September 18 03:35 EDT 2020. Contains 337164 sequences. (Running on oeis4.)