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

1,2

Links

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

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)

Example

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 ...

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 *)

Prog

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

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).

Sequence in context: A024286 A231795 A240319 * A049034 A092546 A227584

Adjacent sequences: A302910 A302911 A302912 * A302914 A302915 A302916

Keyword

nonn

Author

Muniru A Asiru, Apr 15 2018

Status

approved