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.

1, 0, -1, 0, 1, -4, 12, 64, -172, -1348, 3456, 34240, -87084, 370640, -872336, -22639616, 52307088, -181323568, 399580288, 23627011200, -51305628400, -686160247552, 1545932859328, 68098264912128, -155370174372864, 6326621032802304, -13829529077133312, -1087288396552040448

Offset: 0

Author

Stefano Spezia, Apr 19 2022

Keywords

• sign

Examples

			a(8) = -172:
    0,  1,  0,  0,  0,  0,  0,  0;
    1,  0,  1,  2,  0,  0,  0,  0;
    0,  1,  0,  1,  2,  3,  0,  0;
    0,  2,  1,  0,  1,  2,  3,  4;
    0,  0,  2,  1,  0,  1,  2,  3;
    0,  0,  3,  2,  1,  0,  1,  2;
    0,  0,  0,  3,  2,  1,  0,  1;
    0,  0,  0,  4,  3,  2,  1,  0.
		

Crossrefs

Cf. A085750, A085807.

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

Programs

Mathematica

Join[{1},Table[Det[Table[If[Min[i,j]
				

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

Python

from sympy import Matrix
def A353452(n): return Matrix(n, n, lambda i, j: abs(i-j) if min(i,j)Chai Wah Wu, Aug 29 2023

Formula

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

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

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