A355326 Determinant of the n X n matrix [(i-j)^3+d(i,j)]_{1<=i,j<=n}, where d(i,j) is 1 or 0 according as i = j or not.

1, 2, 67, 2157, 96471, 2312410, 32099453, 302049265, 2134677349, 12111035146, 57724828943, 238763085133, 877863236043, 2922096754578, 8932649551321, 25364746314689, 67523106652585, 169800639240178, 405912148130875, 927335183703821, 2033820866612767, 4298718682928682, 8785487346560277, 17412229912018801, 33551232473687501

Offset: 1

Author

Zhi-Wei Sun, Jun 28 2022

Keywords

• nonn

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.

See also A079034 and A355175 for related determinants.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000578, A079034, A355175.

Programs

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}]

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