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

1,2

Links

Table of n, a(n) for n=1..16.

Formula

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

Example

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 ...

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 *)

Prog

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

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.

Sequence in context: A147629 A274256 A280574 * A370576 A174486 A135438

Adjacent sequences: A302907 A302908 A302909 * A302911 A302912 A302913

Keyword

nonn

Author

Muniru A Asiru, Apr 15 2018

Status

approved