A129462 Coefficients of the v=2 member of a family of certain orthogonal polynomials.

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, 36493200, 1753624, 36624, 327, 1

Offset: 0

Author

Wolfdieter Lang, May 04 2007

Keywords

• sign
• tabl
• easy

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.

Examples

			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).
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• 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.

Crossrefs

Columns: A019590 (m=0), A129464 (m=1), A129465 (m=2), A129466 (m=3).

Cf. A000217, A129065 (v=1 triangle), A129463 (row sums).

Programs

Magma

function T(n,k) // T = A129462
  if k lt 0 or k gt n then return 0;
  elif n eq 0 then return 1;
  else return (2*(n-1)*(n-2)-1)*T(n-1,k) - ((n-1)*(n-3))^2*T(n-2,k) + T(n-1,k-1);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 08 2024

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];
T[n_, m_]:= Coefficient[p[n, 2, x], x, m];
Table[T[n, m], {n, 0, 9}, {m, 0, n}]//Flatten (* Jean-François Alcover, Oct 30 2013 *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0, 1, (2*(n-1)*(n-2)- 1)*T[n-1,k] -((n-1)*(n-3))^2*T[n-2,k] +T[n-1,k-1]]]; (* T=A129462 *)
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 08 2024 *)

SageMath

@CachedFunction
def T(n,k): # T = A129462
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return (2*(n-1)*(n-2)-1)*T(n-1,k) - ((n-1)*(n-3))^2*T(n-2,k) + T(n-1,k-1)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 08 2024

Formula

T(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: T(n,m) = T(n-1,m-1) + (2*(n-1)^2 - 2*(v-1)*(n-1) - v + 1)*T(n-1,m) -((n-1)^2*(n-1-v)^2)*T(n-2, m); T(n,m)=0 if n < m, T(-1,m):=0, T(0,0)=1, T(n,-1)=0. Put v=2 for this triangle.

Sum_{k=0..n} T(n, k) = A129463(n) (row sums).