OFFSET
1,2
COMMENTS
Conjecture 1: a(n) = 1 + P(n^2)*n^2*(n^2-1)/672000, where P(n) = n^6 - 19*n^5 + 123*n^4 - 337*n^3 + 12376*n^2 - 44144*n + 40000.
Conjecture 2: For any positive integers m and n, the determinant of the matrix [(i-j)^m+d(i,j)]_{1<=i,j<=n} has the form 1 + n^2*(n^2-1)*P(n), where P(n) is a polynomial in n with rational number coefficients whose degree is (m+1)^2-4.
LINKS
Han Wang and Zhi-Wei Sun, Evaluations of three determinants, arXiv:2206.12317 [math.NT], 2022.
EXAMPLE
a(3) = 67 since the matrix [(i-j)^3+d(i,j)]_{1<=i,j<=3} = [1,-1,-8;1,1,-1;8,1,1] has determinant 67.
MATHEMATICA
a[n_]:=a[n]=Det[Table[If[i==j, 1, (i-j)^3], {i, 1, n}, {j, 1, n}]];
Table[a[n], {n, 1, 25}]
PROG
(PARI) a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, (i-j)^3))); \\ Michel Marcus, Jun 29 2022
(Python)
from sympy import Matrix
def A355326(n): return Matrix(n, n, [1 if i==j else (i-j)**3 for i in range(n) for j in range(n)]).det() # Chai Wah Wu, Jun 29 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 28 2022
STATUS
approved