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A353453 a(n) is the permanent 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 A353453 #18 Aug 29 2023 20:18:26
%S A353453 1,0,1,0,1,4,64,576,7844,63524,882772,11713408,252996564,5879980400,
%T A353453 184839020672,5698866739200,229815005974352,9350598794677712,
%U A353453 480306381374466176,23741710999960266176,1446802666239931811472,86153125248221968292928,6197781268948296566634304
%N A353453 a(n) is the permanent 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 A353453 Sum_{i=1..n+1-k} M[i,i+k] = A173997(n, k) with 1 <= k <= floor((n + 1)/2).
%F A353453 Sum_{i=1..n} Sum_{j=1..n} M[i,j] = 2*A006918(n-1).
%F A353453 Sum_{i=1..n} Sum_{j=1..n} M[i,j]^2 = A350050(n+1).
%e A353453 a(8) = 7844:
%e A353453     0,  1,  0,  0,  0,  0,  0,  0;
%e A353453     1,  0,  1,  2,  0,  0,  0,  0;
%e A353453     0,  1,  0,  1,  2,  3,  0,  0;
%e A353453     0,  2,  1,  0,  1,  2,  3,  4;
%e A353453     0,  0,  2,  1,  0,  1,  2,  3;
%e A353453     0,  0,  3,  2,  1,  0,  1,  2;
%e A353453     0,  0,  0,  3,  2,  1,  0,  1;
%e A353453     0,  0,  0,  4,  3,  2,  1,  0.
%t A353453 Join[{1},Table[Permanent[Table[If[Min[i,j]<Max[i,j]<=2Min[i,j],Abs[j-i],0],{i,n},{j,n}]],{n,22}]]
%o A353453 (PARI) a(n) = matpermanent(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 A353453 (Python)
%o A353453 from sympy import Matrix
%o A353453 def A353453(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).per() if n else 1 # _Chai Wah Wu_, Aug 29 2023
%Y A353453 Cf. A085750, A085807.
%Y A353453 Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353452 (determinant).
%K A353453 nonn
%O A353453 0,6
%A A353453 _Stefano Spezia_, Apr 19 2022