A302911 Determinant of n X n matrix whose main diagonal consists of the first n 7-gonal numbers and all other elements are 1's.

1, 6, 102, 3366, 181764, 14541120, 1614064320, 237267455040, 44606281547520, 10437869882119680, 2974792916404108800, 1014404384493801100800, 407790562566508042521600, 190845983281125763900108800, 102865984988526786742158643200

Offset: 1

Author

Muniru A Asiru, Apr 15 2018

Keywords

• nonn

Examples

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1   7   1   1   1   1   1 ...
  1   1  18   1   1   1   1 ...
  1   1   1  34   1   1   1 ...
  1   1   1   1  55   1   1 ...
  1   1   1   1   1  81   1 ...
  1   1   1   1   1   1 112 ...
		

Crossrefs

Cf. A000566 (heptagonal 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), this sequence (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).

Programs

Maple

d:=(i,j)->`if`(i<>j,1,i*(5*i-3)/2):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..20);

Mathematica

nmax = 20; Table[Det[Table[If[i == j, i*(5*i - 3)/2, 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[5^(n + 1) * Gamma[n] * Gamma[n + 7/5] / (7 * Gamma[2/5] * 2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)

PARI

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(5*i-3)/2))); \\ Michel Marcus, Apr 16 2018

Formula

From Vaclav Kotesovec, Apr 16 2018: (Start)

a(n) = 5^(n + 1) * Gamma(n) * Gamma(n + 7/5) / (7 * Gamma(2/5) * 2^n).

a(n) ~ Pi * 5^(n+1) * n^(2*n + 2/5) / (7 * Gamma(2/5) * 2^(n-1) * exp(2*n)).

a(n+1) = a(n) * n*(5*n + 7)/2.

(End)