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A353452 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.

%I A353452 #19 Aug 30 2023 02:03:13
%S A353452 1,0,-1,0,1,-4,12,64,-172,-1348,3456,34240,-87084,370640,-872336,
%T A353452 -22639616,52307088,-181323568,399580288,23627011200,-51305628400,
%U A353452 -686160247552,1545932859328,68098264912128,-155370174372864,6326621032802304,-13829529077133312,-1087288396552040448
%N A353452 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 A353452 Sum_{i=1..n+1-k} M[i,i+k] = A173997(n, k) with 1 <= k <= floor((n + 1)/2).
%F A353452 Sum_{i=1..n} Sum_{j=1..n} M[i,j] = 2*A006918(n-1).
%F A353452 Sum_{i=1..n} Sum_{j=1..n} M[i,j]^2 = A350050(n+1).
%e A353452 a(8) = -172:
%e A353452     0,  1,  0,  0,  0,  0,  0,  0;
%e A353452     1,  0,  1,  2,  0,  0,  0,  0;
%e A353452     0,  1,  0,  1,  2,  3,  0,  0;
%e A353452     0,  2,  1,  0,  1,  2,  3,  4;
%e A353452     0,  0,  2,  1,  0,  1,  2,  3;
%e A353452     0,  0,  3,  2,  1,  0,  1,  2;
%e A353452     0,  0,  0,  3,  2,  1,  0,  1;
%e A353452     0,  0,  0,  4,  3,  2,  1,  0.
%t A353452 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 A353452 (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 A353452 (Python)
%o A353452 from sympy import Matrix
%o A353452 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 A353452 Cf. A085750, A085807.
%Y A353452 Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353453 (permanent).
%K A353452 sign
%O A353452 0,6
%A A353452 _Stefano Spezia_, Apr 19 2022