A259056 a(n) gives the determinant of a bisymmetric n X n matrix involving the entries 1, 2, ..., A002620(n+1).

1, -3, -16, 60, 384, -1680, -12288, 60480, 491520, -2661120, -23592960, 138378240, 1321205760, -8302694400, -84557168640, 564583219200, 6088116142080, -42908324659200, -487049291366400, 3604299271372800

Offset: 1

Author

Wolfdieter Lang, Jul 07 2015

Keywords

• sign
• easy

Comments

The maximal number of distinct entries of an n X n bisymmetric matrix B_n is A002620(n+1) = ((n+1)/2)^2 if n is odd and = n*(n+2)/4 if n is even. See a comment and example under A002620.

Here the first A002620(n+1) positive numbers are used consecutively, that is B_n[i, j] = (i-1)*n - (i-1)^2 + j for j=i..N-(i-1) and i = 1..ceiling(n/2).

Conjecture: a(2*n-1) = (-1)^(n-1)*A034976(n), n >= 1.

For a(2*n)/3 see A259057(n), n >= 1 (assuming that a(2*n) is always divisible by 3).

Examples

			n = 1, B_1 [1], a(1) = det(B_1) = 1.
n = 2, B_2 = [[1,2],[2,1]], a(2) = det(B_2) = -3.
n = 3: B_3 = [[1,2,3],[2,4,2],[3,2,1]], a(3) = det(B_3) = -16
n = 4: B_4 = [[1,2,3,4],[2,5,6,3],[3,6,5,2],[4,3,2,1]], a(4) = det(B_4) = 60.
		

Links

• Wikipedia, Bisymmetric Matrix .

Crossrefs

Cf. A002620, A259057.

Formula

Determinant of the above defined bisymmetric matrix B_n, n >= 1.