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%I #30 Feb 10 2024 09:24:39
%S 1,-1,1,0,-2,1,0,-6,1,1,0,-48,-4,12,1,0,-720,-204,208,35,1,0,-17280,
%T -7776,5208,1348,74,1,0,-604800,-358560,179688,64580,5138,133,1,0,
%U -29030400,-20839680,8175744,3888528,400384,14952,216,1,0,-1828915200,-1516112640,472666752,291010032,36493200,1753624,36624,327,1
%N Coefficients of the v=2 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. These zeros are from 2*A000217.
%C 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.
%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 The column sequences give A019590, A129464, A129465, A129466 for m=0,1,2,3.
%C 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.
%H G. C. Greubel, <a href="/A129462/b129462.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="/A129462/a129462.txt">First ten rows and more</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=2 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=2 for this triangle.
%F Sum_{k=0..n} T(n, k) = A129463(n) (row sums).
%e Triangle begins:
%e 1;
%e -1, 1;
%e 0, -2, 1;
%e 0, -6, 1, 1;
%e 0, -48, -4, 12, 1;
%e 0, -720, -204, 208, 35, 1;
%e ...
%e Row n=2: [0,-2,1]. p(2,2,x) = x*(x-2).
%e 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).
%t 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 T[n_, m_]:= Coefficient[p[n, 2, x], x, m];
%t Table[T[n, m], {n, 0, 9}, {m, 0, n}]//Flatten (* _Jean-François Alcover_, Oct 30 2013 *)
%t 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 *)
%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 08 2024 *)
%o (Magma)
%o function T(n,k) // T = A129462
%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)*(n-2)-1)*T(n-1,k) - ((n-1)*(n-3))^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 08 2024
%o (SageMath)
%o @CachedFunction
%o def T(n,k): # T = A129462
%o if (k<0 or k>n): return 0
%o elif (n==0): return 1
%o 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)
%o flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Feb 08 2024
%Y Columns: A019590 (m=0), A129464 (m=1), A129465 (m=2), A129466 (m=3).
%Y Cf. A000217, A129065 (v=1 triangle), A129463 (row sums).
%K sign,tabl,easy
%O 0,5
%A _Wolfdieter Lang_, May 04 2007
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