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A129462
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Coefficients of the v=2 member of a family of certain orthogonal polynomials.
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7
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1, -1, 1, 0, -2, 1, 0, -6, 1, 1, 0, -48, -4, 12, 1, 0, -720, -204, 208, 35, 1, 0, -17280, -7776, 5208, 1348, 74, 1, 0, -604800, -358560, 179688, 64580, 5138, 133, 1, 0, -29030400, -20839680, 8175744, 3888528, 400384, 14952, 216, 1, 0, -1828915200, -1516112640, 472666752, 291010032, 36493200, 1753624, 36624, 327, 1
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OFFSET
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0,5
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COMMENTS
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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.
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.
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=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.
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LINKS
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FORMULA
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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.
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.
Sum_{k=0..n} T(n, k) = A129463(n) (row sums).
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EXAMPLE
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Triangle begins:
1;
-1, 1;
0, -2, 1;
0, -6, 1, 1;
0, -48, -4, 12, 1;
0, -720, -204, 208, 35, 1;
...
Row n=2: [0,-2,1]. p(2,2,x) = x*(x-2).
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).
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MATHEMATICA
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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[n_, m_]:= Coefficient[p[n, 2, x], x, m];
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 *)
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 08 2024 *)
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PROG
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(Magma)
if k lt 0 or k gt n then return 0;
elif n eq 0 then 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);
end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 08 2024
(SageMath)
@CachedFunction
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)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 08 2024
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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