A129065 Coefficients of the v=1 member of a family of certain orthogonal polynomials.

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

Offset: 0

Author

Wolfdieter Lang, May 04 2007

Keywords

• nonn
• 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.

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.

Examples

			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]].
		

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.

Crossrefs

Columns: A000007 (m=0), A010790, (m=1), A129460 (m=2), A129461 (m=3).

Cf. A129458 (row sums), A129462 (v=2 triangle).

Programs

Magma

function T(n,k) // T = A129065
  if k lt 0 or k gt n then return 0;
  elif n eq 0 then return 1;
  else return 2*(n-1)^2*T(n-1,k) - 4*Binomial(n-1,2)^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 07 2024

Mathematica

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

SageMath

@CachedFunction
def T(n,k): # T = A129065
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return 2*(n-1)^2*T(n-1,k) - 4*binomial(n-1,2)^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 07 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=1 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=1 here.

Sum_{k=0..n} T(n, k) = A129458(n) (row sums).