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A302910 Determinant of n X n matrix whose main diagonal consists of the first n 6-gonal numbers and all other elements are 1's.

%I A302910 #27 Mar 23 2020 18:34:26
%S A302910 1,5,70,1890,83160,5405400,486486000,57891834000,8799558768000,
%T A302910 1663116607152000,382516819644960000,105192125402364000000,
%U A302910 34082248630365936000000,12849007733647957872000000,5576469356403213716448000000,2760352331419590789641760000000
%N A302910 Determinant of n X n matrix whose main diagonal consists of the first n 6-gonal numbers and all other elements are 1's.
%F A302910 a(n) = (n + 1/2) * (2*n-1)! / (3 * 2^(n-2)). - _Vaclav Kotesovec_, Apr 16 2018
%e A302910 The matrix begins:
%e A302910   1  1  1  1  1  1  1 ...
%e A302910   1  6  1  1  1  1  1 ...
%e A302910   1  1 15  1  1  1  1 ...
%e A302910   1  1  1 28  1  1  1 ...
%e A302910   1  1  1  1 45  1  1 ...
%e A302910   1  1  1  1  1 66  1 ...
%e A302910   1  1  1  1  1  1 91 ...
%p A302910 d:=(i,j)->`if`(i<>j,1,i*(2*i-1)):
%p A302910 seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..20);
%t A302910 nmax = 20; Table[Det[Table[If[i == j, i*(2*i - 1), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* _Vaclav Kotesovec_, Apr 16 2018 *)
%t A302910 Table[(n + 1/2) * (2*n - 1)! / (3 * 2^(n - 2)), {n, 1, 20}] (* _Vaclav Kotesovec_, Apr 16 2018 *)
%t A302910 Table[Det[DiagonalMatrix[PolygonalNumber[6,Range[n]]]/.(0->1)],{n,20}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 23 2020 *)
%o A302910 (PARI) a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(2*i-1)))); \\ _Michel Marcus_, Apr 16 2018
%Y A302910 Cf. A000384 (hexagonal numbers).
%Y A302910 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), this sequence (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).
%Y A302910 Odd bisection of column k=1 of A097591.
%K A302910 nonn
%O A302910 1,2
%A A302910 _Muniru A Asiru_, Apr 15 2018