A129458 -id:A129458 - cp's OEIS frontend

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

Wolfdieter Lang, May 04 2007

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.

			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]].

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.

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

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

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

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 *)

@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

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).

A129463 Row sums of triangle A129462 (v=2 member of a certain family).

Original entry on oeis.org

1, 0, -1, -4, -39, -680, -18425, -713820, -37390255, -2543067280, -217799766225, -22928327328500, -2909576503498775, -437960283393276600, -77145678498655849225, -15720035935018890359500, -3668950667796545284209375, -972327797466833893742228000

Offset: 0

Wolfdieter Lang, May 04 2007

See A129462 for the M. Bruschi et al. reference.

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A129458 (row sums v=1 member), A129462.

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 *)
A129463[n_]:= A129463[n]= Sum[T[n,k], {k,0,n}];
Table[A129463[n], {n,0,40}] (* G. C. Greubel, Feb 08 2024 *)

@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)
def A129463(n): return sum(T(n,k) for k in range(n+1))
[A129463(n) for n in range(41)] # G. C. Greubel, Feb 08 2024

a(n) = Sum_{k=0..n} A129462(n,k), n >= 0.
From Vaclav Kotesovec, Aug 24 2016: (Start)
a(n) = 2*(n-2)*(n-1)*a(n-1) - (n-3)^2*(n-1)^2*a(n-2).
a(n) ~ c * n^(2*n+(sqrt(5)-3)/2) / exp(2*n), where c = -2.3203776630375605070105975273368548459...
(End)

cp's OEIS Frontend

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

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