cp's OEIS Frontend

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.

%I A355175 #31 Jun 29 2022 10:15:48
%S A355175 1,0,-9,45,931,6392,29205,104497,315469,838696,2018523,4482765,
%T A355175 9314943,18301648,34277321,61592769,106738105,179155504,292282207,
%U A355175 464869581,722629755,1100267400,1643960605,2414361521,3490194501,4972536856,6989875075,9704037421,13317112215,18079469856,24299015697,32351810305,42694203377,55876637664
%N 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.
%C A355175 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.
%C A355175 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.
%H A355175 Han Wang and Zhi-Wei Sun, <a href="http://arxiv.org/abs/2206.12317">Evaluations of three determinants</a>, arXiv:2206.12317 [math.NT], 2022.
%e A355175 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.
%t A355175 a[n_]:=a[n]=Det[Table[If[i==j,1,(i-j)^2],{i,1,n},{j,1,n}]]
%t A355175 Table[a[n],{n,1,34}]
%o A355175 (Python)
%o A355175 from sympy import Matrix
%o A355175 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
%o A355175 (PARI) a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, (i-j)^2))); \\ _Michel Marcus_, Jun 29 2022
%Y A355175 Cf. A000290, A079034.
%K A355175 sign
%O A355175 1,3
%A A355175 _Zhi-Wei Sun_, Jun 28 2022