A129462 Coefficients of the v=2 member of a family of certain orthogonal polynomials.
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
Offset: 0
Examples
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).
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 and more.
Crossrefs
Programs
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Magma
function T(n,k) // T = A129462 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
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Mathematica
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]; Table[T[n, m], {n, 0, 9}, {m, 0, n}]//Flatten (* Jean-François Alcover, Oct 30 2013 *) 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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SageMath
@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) flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 08 2024
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=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).
Comments