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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.
4
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
0,6
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).
EXAMPLE
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.
MATHEMATICA
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}]]
PROG
(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)<max(i, j)<=(min(i, j)<<1)+1 else 0).det() # Chai Wah Wu, Aug 29 2023
CROSSREFS
Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353453 (permanent).
Sequence in context: A291772 A222645 A259816 * A071769 A221667 A275527
KEYWORD
sign
AUTHOR
Stefano Spezia, Apr 19 2022
STATUS
approved