%I #33 Feb 11 2024 02:48:09

%S 1,0,1,0,2,1,0,12,10,1,0,144,156,28,1,0,2880,3696,908,60,1,0,86400,

%T 125280,37896,3508,110,1,0,3628800,5780160,2036592,236472,10528,182,1,

%U 0,203212800,349090560,138517632,19022736,1074176,26600,280,1

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

%C For v >= 1 the orthogonal polynomials p(n,v,x) have v integer zeros k*(k-1), k = 1..v, for every n >= v.

%C 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.

%C 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.

%C 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.

%C The column sequences give A000007, A010790, A129460, A129461 for m=0,1,2,3.

%H G. C. Greubel, <a href="/A129065/b129065.txt">Rows n = 0..50 of the triangle, flattened</a>

%H M. Bruschi, F. Calogero and R. Droghei, <a href="http://dx.doi.org/10.1088/1751-8113/40/14/005">Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials</a>, J. Physics A, 40(2007), pp. 3815-3829.

%H Wolfdieter Lang, <a href="/A129065/a129065.txt">First ten rows</a>.

%F 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.

%F 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.

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

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 2, 1;

%e 0, 12, 10, 1;

%e 0, 144, 156, 28, 1;

%e 0, 2880, 3696, 908, 60, 1;

%e ...

%e 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.

%e 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]].

%t 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 *)

%o (Magma)

%o function T(n,k) // T = A129065

%o if k lt 0 or k gt n then return 0;

%o elif n eq 0 then return 1;

%o 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);

%o end if;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 07 2024

%o (SageMath)

%o @CachedFunction

%o def T(n,k): # T = A129065

%o if (k<0 or k>n): return 0

%o elif (n==0): return 1

%o 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)

%o flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Feb 07 2024

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

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

%K nonn,tabl,easy

%O 0,5

%A _Wolfdieter Lang_, May 04 2007