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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.

%I A302912 #25 Apr 18 2018 03:31:02
%S A302912 1,7,140,5460,349440,33196800,4381977600,766846080000,171773521920000,
%T A302912 47924812615680000,16294436289331200000,6631835569757798400000,
%U A302912 3183281073483743232000000,1779454120077412466688000000,1145968453329853628547072000000
%N 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.
%F A302912 From _Vaclav Kotesovec_, Apr 16 2018: (Start)
%F A302912 a(n) = 3^(n+1) * Gamma(n) * Gamma(n + 4/3) / (4*Gamma(1/3)).
%F A302912 a(n) ~ Pi * 3^(n+1) * n^(2*n + 1/3) / (2 * Gamma(1/3) * exp(2*n)).
%F A302912 a(n+1) = a(n) * n*(3*n + 4).
%F A302912 (End)
%e A302912 The matrix begins:
%e A302912   1   1   1   1   1   1   1 ...
%e A302912   1   8   1   1   1   1   1 ...
%e A302912   1   1  21   1   1   1   1 ...
%e A302912   1   1   1  40   1   1   1 ...
%e A302912   1   1   1   1  65   1   1 ...
%e A302912   1   1   1   1   1  96   1 ...
%e A302912   1   1   1   1   1   1 133 ...
%p A302912 d:=(i,j)->`if`(i<>j,1,i*(3*i-2)):
%p A302912 seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);
%t A302912 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 *)
%t A302912 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 *)
%o A302912 (PARI) a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(3*i-2)))); \\ _Michel Marcus_, Apr 16 2018
%Y A302912 Cf. A000567 (octagonal numbers).
%Y A302912 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).
%K A302912 nonn
%O A302912 1,2
%A A302912 _Muniru A Asiru_, Apr 15 2018