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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. 5
1, 8, 184, 8280, 612720, 67399200, 10312077600, 2093351752800, 544271455728000, 176343951655872000, 69655860904069440000, 32947222207624845120000, 18384549991854663576960000, 11949957494705531325024000000, 8950518163534442962442976000000 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 19 20:54 EST 2019. Contains 320328 sequences. (Running on oeis4.)