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A372409
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Determinant of the matrix obtained from [Jacobi(i-j, 2*n+1)]_{0<i,j<n} by replacing all the entries in the first row by 1, where Jacobi(a,m) denotes the Jacobi symbol (a/m).
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1
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1, -1, 1, -3, 3, -8, -5, -5, -24, 7, 0, 0, 9, 9, -81312, -1341867, 11, -19685120, -13, -13, 0, -15, 0, -180287762432, 17, -1407939911477, 10526233598464, 19, 19, 0, 6040299856799, -21, 29830847001120768, 23, -23, 0, 115407361849089836, -25, 0, 27, 104060523591574200
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OFFSET
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2,4
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COMMENTS
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Conjecture 1: If 2*n + 1 (with n > 1) is a prime p, then a(n)/floor((p-2)/3) coincides with (-1)^((p+3)/4) if p == 1 (mod 4), and (-1)^((h(-p)+1)/2) if p == 3 (mod 4), where h(-p) is the class number of the imaginary quadratic field Q(sqrt(-p)).
Conjecture 2: Let p > 3 be a prime, and let S(p) and T(p) denote the matrices obtained from [Jacobi(i+j,p)]_{1<=i,j<=(p-3)/2} and [Jacobi(i+j,p)]_{0<=i,j<=(p-3)/2} (respectively) by replacing all the entries in the first row by 1. Then det S(p) = -det T(p) = 2^((p-5)/2)*s(p), where s(p) is (-1)^((p+3)/4) if p == 1 (mod 4), and (-1)^((h(-p)-1)/2) if p == 3 (mod 4).
Both conjectures are motivated by Conjecture 4.6 in the author's 2019 FFA paper as well as the conjectures in A372385. They have been verified for primes p < 2000.
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REFERENCES
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L.-Y. Wang and H.-L. Wu, On certain determinants involving Legendre symbols, Ramanujan J. 58 (2022), 43-56.
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LINKS
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EXAMPLE
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a(3) = -1 since the determinant of the matrix [1, 1; Jacobi(2-1,2*3+1), Jacobi(2-2,2*3+1)] = [1, 1; 1, 0] has the value -1.
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MATHEMATICA
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a[n_]:=a[n]=Det[Table[If[i==1, 1, JacobiSymbol[i-j, 2*n+1]], {i, 1, n-1}, {j, 1, n-1}]];
tab={}; Do[tab=Append[tab, a[n]], {n, 2, 42}]; Print[tab]
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PROG
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(PARI) a(n) = matdet(matrix(n-1, n-1, i, j, if (i==1, 1, kronecker(i-j, 2*n+1)))); \\ Michel Marcus, Apr 30 2024
(Python)
from sympy import Matrix, jacobi_symbol
def A372409(n): return Matrix(n-1, n-1, [jacobi_symbol(i-j, (n<<1)|1) if i else 1 for i in range(n-1) for j in range(n-1)]).det() # Chai Wah Wu, May 01 2024
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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