%I #19 Aug 30 2023 02:03:13

%S 1,0,-1,0,1,-4,12,64,-172,-1348,3456,34240,-87084,370640,-872336,

%T -22639616,52307088,-181323568,399580288,23627011200,-51305628400,

%U -686160247552,1545932859328,68098264912128,-155370174372864,6326621032802304,-13829529077133312,-1087288396552040448

%N a(n) is the determinant of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

%F Sum_{i=1..n+1-k} M[i,i+k] = A173997(n, k) with 1 <= k <= floor((n + 1)/2).

%F Sum_{i=1..n} Sum_{j=1..n} M[i,j] = 2*A006918(n-1).

%F Sum_{i=1..n} Sum_{j=1..n} M[i,j]^2 = A350050(n+1).

%e a(8) = -172:

%e 0, 1, 0, 0, 0, 0, 0, 0;

%e 1, 0, 1, 2, 0, 0, 0, 0;

%e 0, 1, 0, 1, 2, 3, 0, 0;

%e 0, 2, 1, 0, 1, 2, 3, 4;

%e 0, 0, 2, 1, 0, 1, 2, 3;

%e 0, 0, 3, 2, 1, 0, 1, 2;

%e 0, 0, 0, 3, 2, 1, 0, 1;

%e 0, 0, 0, 4, 3, 2, 1, 0.

%t Join[{1},Table[Det[Table[If[Min[i,j]<Max[i,j]<=2Min[i,j],Abs[j-i],0],{i,n},{j,n}]],{n,27}]]

%o (PARI) a(n) = matdet(matrix(n, n, i, j, if ((min(i,j) < max(i,j)) && (max(i,j) <= 2*min(i,j)), abs(i-j)))); \\ _Michel Marcus_, Apr 20 2022

%o (Python)

%o from sympy import Matrix

%o def A353452(n): return Matrix(n, n, lambda i, j: abs(i-j) if min(i,j)<max(i,j)<=(min(i,j)<<1)+1 else 0).det() # _Chai Wah Wu_, Aug 29 2023

%Y Cf. A085750, A085807.

%Y Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353453 (permanent).

%K sign

%O 0,6

%A _Stefano Spezia_, Apr 19 2022