A302910 Determinant of n X n matrix whose main diagonal consists of the first n 6-gonal numbers and all other elements are 1's.

1, 5, 70, 1890, 83160, 5405400, 486486000, 57891834000, 8799558768000, 1663116607152000, 382516819644960000, 105192125402364000000, 34082248630365936000000, 12849007733647957872000000, 5576469356403213716448000000, 2760352331419590789641760000000

Offset: 1

Author

Muniru A Asiru, Apr 15 2018

Keywords

• nonn

Examples

			The matrix begins:
  1  1  1  1  1  1  1 ...
  1  6  1  1  1  1  1 ...
  1  1 15  1  1  1  1 ...
  1  1  1 28  1  1  1 ...
  1  1  1  1 45  1  1 ...
  1  1  1  1  1 66  1 ...
  1  1  1  1  1  1 91 ...
		

Crossrefs

Cf. A000384 (hexagonal numbers).

Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), this sequence (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).

Odd bisection of column k=1 of A097591.

Programs

Maple

d:=(i,j)->`if`(i<>j,1,i*(2*i-1)):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..20);

Mathematica

nmax = 20; Table[Det[Table[If[i == j, i*(2*i - 1), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[(n + 1/2) * (2*n - 1)! / (3 * 2^(n - 2)), {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[Det[DiagonalMatrix[PolygonalNumber[6,Range[n]]]/.(0->1)],{n,20}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 23 2020 *)

PARI

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(2*i-1)))); \\ Michel Marcus, Apr 16 2018

Formula

a(n) = (n + 1/2) * (2*n-1)! / (3 * 2^(n-2)). - Vaclav Kotesovec, Apr 16 2018