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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.
4
1, 0, 1, 0, 1, 4, 64, 576, 7844, 63524, 882772, 11713408, 252996564, 5879980400, 184839020672, 5698866739200, 229815005974352, 9350598794677712, 480306381374466176, 23741710999960266176, 1446802666239931811472, 86153125248221968292928, 6197781268948296566634304
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) = 7844:
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[Permanent[Table[If[Min[i, j]<Max[i, j]<=2Min[i, j], Abs[j-i], 0], {i, n}, {j, n}]], {n, 22}]]
PROG
(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
(Python)
from sympy import Matrix
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
CROSSREFS
Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353452 (determinant).
Sequence in context: A195800 A224446 A128782 * A318779 A264567 A264575
KEYWORD
nonn
AUTHOR
Stefano Spezia, Apr 19 2022
STATUS
approved