A353453 - OEIS

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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.

1, 0, 1, 0, 1, 4, 64, 576, 7844, 63524, 882772, 11713408, 252996564, 5879980400, 184839020672, 5698866739200, 229815005974352, 9350598794677712, 480306381374466176, 23741710999960266176, 1446802666239931811472, 86153125248221968292928, 6197781268948296566634304

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..22.

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. A085750, A085807.

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

Adjacent sequences: A353450 A353451 A353452 * A353454 A353455 A353456

KEYWORD

nonn

AUTHOR

Stefano Spezia, Apr 19 2022

STATUS

approved