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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. 5
1, 7, 140, 5460, 349440, 33196800, 4381977600, 766846080000, 171773521920000, 47924812615680000, 16294436289331200000, 6631835569757798400000, 3183281073483743232000000, 1779454120077412466688000000, 1145968453329853628547072000000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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
KEYWORD
nonn
AUTHOR
Muniru A Asiru, Apr 15 2018
STATUS
approved

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Last modified July 14 20:04 EDT 2024. Contains 374323 sequences. (Running on oeis4.)