A302914 Determinant of n X n matrix whose main diagonal consists of the first n 10-gonal numbers and all other elements are 1's.

1, 9, 234, 11934, 1002456, 125307000, 21803418000, 5036589558000, 1490830509168000, 550116457882992000, 247552406047346400000, 133430746859519709600000, 84861955002654535305600000, 62882708656967010661449600000, 53701833193049827104877958400000

Offset

1,2

Comments

From Vaclav Kotesovec, Apr 16 2018: (Start)

In general, for k > 2, these determinants for k-gonal numbers satisfies:

a(n,k) = ((k-2)/2)^(n-1) * Gamma(n) * Gamma(n + k/(k-2)) / Gamma(1 + k/(k-2)).

a(n,k) ~ 4*Pi * (k/2 - 1)^n * n^(2*n + 2/(k-2)) / (k * Gamma(k/(k-2)) * exp(2*n)).

a(n+1,k) = a(n,k) * n*((k-2)*n + k)/2.

(End)

Links

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

Formula

From Vaclav Kotesovec, Apr 16 2018: (Start)

a(n) = 4^(n+1) * Gamma(n) * Gamma(n + 5/4) / (5*Gamma(1/4)).

a(n) ~ Pi * 2^(2*n + 3) * n^(2*n + 1/4) / (5 * Gamma(1/4) * exp(2*n)).

a(n+1) = a(n) * n*(4*n + 5).

(End)

Example

The matrix begins:
  1   1   1   1   1   1   1 ...
  1  10   1   1   1   1   1 ...
  1   1  27   1   1   1   1 ...
  1   1   1  52   1   1   1 ...
  1   1   1   1  85   1   1 ...
  1   1   1   1   1 126   1 ...
  1   1   1   1   1   1 175 ...

Maple

d:=(i, j)->`if`(i<>j, 1, i*(4*i-3)):
seq(LinearAlgebra[Determinant](Matrix(n, d)), n=1..16);

Mathematica

nmax = 20; Table[Det[Table[If[i == j, i*(4*i-3), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
RecurrenceTable[{a[n+1] == a[n] * n*(4*n + 5), a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[4^(n+1) * Gamma[n] * Gamma[n + 5/4] / (5*Gamma[1/4])], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)

Prog

(PARI) a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(4*i-3)))); \\ Michel Marcus, Apr 16 2018

Crossrefs

Cf. A001107.

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), A302913 (k=9), this sequence (k=10).

Cf. A007840 (permanent instead of determinant, for k=2).

Sequence in context: A153223 A276536 A288683 * A157569 A279492 A297747

Adjacent sequences: A302911 A302912 A302913 * A302915 A302916 A302917

Keyword

nonn

Author

Muniru A Asiru, Apr 15 2018

Status

approved