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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.

%I A355326 #17 Jun 29 2022 13:41:07
%S A355326 1,2,67,2157,96471,2312410,32099453,302049265,2134677349,12111035146,
%T A355326 57724828943,238763085133,877863236043,2922096754578,8932649551321,
%U A355326 25364746314689,67523106652585,169800639240178,405912148130875,927335183703821,2033820866612767,4298718682928682,8785487346560277,17412229912018801,33551232473687501
%N 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.
%C A355326 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.
%C A355326 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.
%C A355326 See also A079034 and A355175 for related determinants.
%H A355326 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 A355326 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.
%t A355326 a[n_]:=a[n]=Det[Table[If[i==j,1,(i-j)^3],{i,1,n},{j,1,n}]];
%t A355326 Table[a[n],{n,1,25}]
%o A355326 (PARI) a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, (i-j)^3))); \\ _Michel Marcus_, Jun 29 2022
%o A355326 (Python)
%o A355326 from sympy import Matrix
%o A355326 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
%Y A355326 Cf. A000578, A079034, A355175.
%K A355326 nonn
%O A355326 1,2
%A A355326 _Zhi-Wei Sun_, Jun 28 2022