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

0,5

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

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.

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).

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; 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 *)

Prog

(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
(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

Crossrefs

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

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

Sequence in context: A039910 A352399 A129467 * A361718 A355565 A202700

Adjacent sequences: A129062 A129063 A129064 * A129066 A129067 A129068

Keyword

nonn,tabl,easy

Author

Wolfdieter Lang, May 04 2007

Status

approved