A355175 Determinant of the n X n matrix [(i-j)^2 + d(i,j)]_{1<=i,j<=n}, where d(i,j) is 1 or 0 according as i = j or not.

1, 0, -9, 45, 931, 6392, 29205, 104497, 315469, 838696, 2018523, 4482765, 9314943, 18301648, 34277321, 61592769, 106738105, 179155504, 292282207, 464869581, 722629755, 1100267400, 1643960605, 2414361521, 3490194501, 4972536856, 6989875075, 9704037421, 13317112215, 18079469856, 24299015697, 32351810305, 42694203377, 55876637664

Offset: 1

Author

Zhi-Wei Sun, Jun 28 2022

Keywords

• sign

Comments

Conjecture: a(n) = (n^2-4)*(n^2+2*n+3)*(n^5-2*n^4-n^3-28*n^2+60*n-90)/1080.

In 2022, Han Wang and Zhi-Wei Sun determined the values of det[i-j+d(i,j)]{1<=i,j<=n} and det[|i-j|+d(i,j)]{1<=i,j<=n}, where d(i,j) is 1 or 0 according as i = j or not.

Examples

			a(3) = -9 since the matrix [(i-j)^2+d(i,j)]_{1<=i,j<=3} = [1,1,4; 1,1,1; 4,1,1] has determinant -9.
		

Links

• Han Wang and Zhi-Wei Sun, Evaluations of three determinants, arXiv:2206.12317 [math.NT], 2022.

Crossrefs

Cf. A000290, A079034.

Programs

Mathematica

a[n_]:=a[n]=Det[Table[If[i==j,1,(i-j)^2],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,34}]

PARI

a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, (i-j)^2))); \\ Michel Marcus, Jun 29 2022

Python

from sympy import Matrix
def A355175(n): return Matrix(n,n,[(i-j)**2 + int(i==j) for i in range(n) for j in range(n)]).det() # Chai Wah Wu, Jun 28 2022