A302913 Determinant of n X n matrix whose main diagonal consists of the first n 9-gonal numbers and all other elements are 1's.

1, 8, 184, 8280, 612720, 67399200, 10312077600, 2093351752800, 544271455728000, 176343951655872000, 69655860904069440000, 32947222207624845120000, 18384549991854663576960000, 11949957494705531325024000000, 8950518163534442962442976000000

Offset: 1

Author

Muniru A Asiru, Apr 15 2018

Keywords

• nonn

Examples

			The matrix begins:
1   1   1   1   1   1   1 ...
1   9   1   1   1   1   1 ...
1   1  24   1   1   1   1 ...
1   1   1  46   1   1   1 ...
1   1   1   1  75   1   1 ...
1   1   1   1   1 111   1 ...
1   1   1   1   1   1 154 ...
		

Crossrefs

Cf. A001106 (nonagonal 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), A302912 (k=8), this sequence (k=9), A302914 (k=10).

Programs

Maple

d:=(i,j)->`if`(i<>j,1,i*(7*i-5)/2):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);

Mathematica

nmax = 20; Table[Det[Table[If[i == j, i*(7*i-5)/2, 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
RecurrenceTable[{a[n+1] == a[n] * n*(7*n + 9)/2, a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[7^(n + 1) * Gamma[n] * Gamma[n + 9/7] / (9*Gamma[2/7]*2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)

PARI

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(7*i-5)/2))); \\ Michel Marcus, Apr 16 2018

Formula

From Vaclav Kotesovec, Apr 16 2018: (Start)

a(n) = 7^(n+1) * Gamma(n) * Gamma(n + 9/7) / (9 * Gamma(2/7) * 2^n).

a(n) ~ Pi * 7^(n+1) * n^(2*n + 2/7) / (9 * Gamma(2/7) * 2^(n-1) * exp(2*n)).

a(n+1) = a(n) * n*(7*n + 9)/2.

(End)