A302912 Determinant of n X n matrix whose main diagonal consists of the first n 8-gonal numbers and all other elements are 1's.

1, 7, 140, 5460, 349440, 33196800, 4381977600, 766846080000, 171773521920000, 47924812615680000, 16294436289331200000, 6631835569757798400000, 3183281073483743232000000, 1779454120077412466688000000, 1145968453329853628547072000000

Offset

1,2

Links

Table of n, a(n) for n=1..15.

Formula

From Vaclav Kotesovec, Apr 16 2018: (Start)

a(n) = 3^(n+1) * Gamma(n) * Gamma(n + 4/3) / (4*Gamma(1/3)).

a(n) ~ Pi * 3^(n+1) * n^(2*n + 1/3) / (2 * Gamma(1/3) * exp(2*n)).

a(n+1) = a(n) * n*(3*n + 4).

(End)

Example

The matrix begins:
  1   1   1   1   1   1   1 ...
  1   8   1   1   1   1   1 ...
  1   1  21   1   1   1   1 ...
  1   1   1  40   1   1   1 ...
  1   1   1   1  65   1   1 ...
  1   1   1   1   1  96   1 ...
  1   1   1   1   1   1 133 ...

Maple

d:=(i, j)->`if`(i<>j, 1, i*(3*i-2)):
seq(LinearAlgebra[Determinant](Matrix(n, d)), n=1..16);

Mathematica

nmax = 20; Table[Det[Table[If[i == j, i*(3*i - 2), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[3^(n+1) * Gamma[n] * Gamma[n + 4/3] / (4*Gamma[1/3])], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)RecurrenceTable[{a[n+1] == a[n] * n * (3*n + 4), a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)

Prog

(PARI) a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(3*i-2)))); \\ Michel Marcus, Apr 16 2018

Crossrefs

Cf. A000567 (octagonal 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), A302910 (k=6), A302911 (k=7), this sequence (k=8), A302913 (k=9), A302914 (k=10).

Sequence in context: A054606 A351280 A306628 * A191956 A215042 A221267

Adjacent sequences: A302909 A302910 A302911 * A302913 A302914 A302915

Keyword

nonn

Author

Muniru A Asiru, Apr 15 2018

Status

approved