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A302909 Determinant of n X n matrix whose main diagonal consists of the first n 5-gonal numbers and all other elements are 1's.

%I A302909 #32 Feb 07 2025 19:43:04
%S A302909 1,4,44,924,31416,1570800,108385200,9863053200,1144114171200,
%T A302909 164752440652800,28831677114240000,6025820516876160000,
%U A302909 1482351847151535360000,423952628285339112960000,139480414705876568163840000,52305155514703713061440000000
%N A302909 Determinant of n X n matrix whose main diagonal consists of the first n 5-gonal numbers and all other elements are 1's.
%H A302909 Muniru A Asiru, <a href="/A302909/b302909.txt">Table of n, a(n) for n = 1..100</a>
%F A302909 From _Vaclav Kotesovec_, Apr 16 2018: (Start)
%F A302909 a(n) = Gamma(n) * Gamma(n + 5/3) * 3^(n + 1) / (5 * Gamma(2/3) * 2^n).
%F A302909 a(n) ~ Gamma(1/3) * 3^(n + 3/2) * n^(2*n + 2/3) / (5 * 2^n * exp(2*n)).
%F A302909 (End)
%F A302909 a(n + 1) = A115067(n + 1) * a(n) = a(n) * n*(3*n + 5)/2. - _David A. Corneth_, Apr 16 2018
%e A302909 The 7 X 7 matrix (as below) has determinant 108385200.
%e A302909   1  1  1  1  1  1  1
%e A302909   1  5  1  1  1  1  1
%e A302909   1  1 12  1  1  1  1
%e A302909   1  1  1 22  1  1  1
%e A302909   1  1  1  1 35  1  1
%e A302909   1  1  1  1  1 51  1
%e A302909   1  1  1  1  1  1 70
%p A302909 d:=(i,j)->`if`(i<>j,1,i*(3*i-1)/2):
%p A302909 seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..17);
%t A302909 Table[FullSimplify[Gamma[n] * Gamma[n + 5/3] * 3^(n + 1) / (5 * Gamma[2/3] * 2^n)], {n, 1, 15}] (* _Vaclav Kotesovec_, Apr 16 2018 *)
%t A302909 Module[{nn=20,pn5},pn5=PolygonalNumber[5,Range[nn]];Table[Det[DiagonalMatrix[Take[pn5,n]]/.(0->1)],{n,nn}]] (* _Harvey P. Dale_, Feb 07 2025 *)
%o A302909 (PARI) a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(3*i-1)/2))); \\ _Michel Marcus_, Apr 16 2018
%o A302909 (PARI) first(n) = my(res = vector(n)); res[1] = 1; for(i = 1, n - 1, res[i + 1] = res[i] * i*(3*i + 5)/2); res \\ _David A. Corneth_, Apr 16 2018
%Y A302909 Cf. A000326, A115067.
%Y A302909 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), this sequence (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).
%K A302909 nonn
%O A302909 1,2
%A A302909 _Muniru A Asiru_, Apr 15 2018