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%I #25 Apr 18 2018 03:31:13
%S 1,8,184,8280,612720,67399200,10312077600,2093351752800,
%T 544271455728000,176343951655872000,69655860904069440000,
%U 32947222207624845120000,18384549991854663576960000,11949957494705531325024000000,8950518163534442962442976000000
%N Determinant of n X n matrix whose main diagonal consists of the first n 9-gonal numbers and all other elements are 1's.
%F From _Vaclav Kotesovec_, Apr 16 2018: (Start)
%F a(n) = 7^(n+1) * Gamma(n) * Gamma(n + 9/7) / (9 * Gamma(2/7) * 2^n).
%F a(n) ~ Pi * 7^(n+1) * n^(2*n + 2/7) / (9 * Gamma(2/7) * 2^(n-1) * exp(2*n)).
%F a(n+1) = a(n) * n*(7*n + 9)/2.
%F (End)
%e The matrix begins:
%e 1 1 1 1 1 1 1 ...
%e 1 9 1 1 1 1 1 ...
%e 1 1 24 1 1 1 1 ...
%e 1 1 1 46 1 1 1 ...
%e 1 1 1 1 75 1 1 ...
%e 1 1 1 1 1 111 1 ...
%e 1 1 1 1 1 1 154 ...
%p d:=(i,j)->`if`(i<>j,1,i*(7*i-5)/2):
%p seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);
%t 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 *)
%t RecurrenceTable[{a[n+1] == a[n] * n*(7*n + 9)/2, a[1] == 1}, a, {n, 1, 20}] (* _Vaclav Kotesovec_, Apr 16 2018 *)
%t 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 *)
%o (PARI) a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(7*i-5)/2))); \\ _Michel Marcus_, Apr 16 2018
%Y Cf. A001106 (nonagonal numbers).
%Y 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).
%K nonn
%O 1,2
%A _Muniru A Asiru_, Apr 15 2018
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