This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A302914 #30 Apr 18 2018 03:31:40
%S A302914 1,9,234,11934,1002456,125307000,21803418000,5036589558000,
%T A302914 1490830509168000,550116457882992000,247552406047346400000,
%U A302914 133430746859519709600000,84861955002654535305600000,62882708656967010661449600000,53701833193049827104877958400000
%N 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.
%C A302914 From _Vaclav Kotesovec_, Apr 16 2018: (Start)
%C A302914 In general, for k > 2, these determinants for k-gonal numbers satisfies:
%C A302914 a(n,k) = ((k-2)/2)^(n-1) * Gamma(n) * Gamma(n + k/(k-2)) / Gamma(1 + k/(k-2)).
%C A302914 a(n,k) ~ 4*Pi * (k/2 - 1)^n * n^(2*n + 2/(k-2)) / (k * Gamma(k/(k-2)) * exp(2*n)).
%C A302914 a(n+1,k) = a(n,k) * n*((k-2)*n + k)/2.
%C A302914 (End)
%F A302914 From _Vaclav Kotesovec_, Apr 16 2018: (Start)
%F A302914 a(n) = 4^(n+1) * Gamma(n) * Gamma(n + 5/4) / (5*Gamma(1/4)).
%F A302914 a(n) ~ Pi * 2^(2*n + 3) * n^(2*n + 1/4) / (5 * Gamma(1/4) * exp(2*n)).
%F A302914 a(n+1) = a(n) * n*(4*n + 5).
%F A302914 (End)
%e A302914 The matrix begins:
%e A302914 1 1 1 1 1 1 1 ...
%e A302914 1 10 1 1 1 1 1 ...
%e A302914 1 1 27 1 1 1 1 ...
%e A302914 1 1 1 52 1 1 1 ...
%e A302914 1 1 1 1 85 1 1 ...
%e A302914 1 1 1 1 1 126 1 ...
%e A302914 1 1 1 1 1 1 175 ...
%p A302914 d:=(i,j)->`if`(i<>j,1,i*(4*i-3)):
%p A302914 seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);
%t A302914 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 *)
%t A302914 RecurrenceTable[{a[n+1] == a[n] * n*(4*n + 5), a[1] == 1}, a, {n, 1, 20}] (* _Vaclav Kotesovec_, Apr 16 2018 *)
%t A302914 Table[FullSimplify[4^(n+1) * Gamma[n] * Gamma[n + 5/4] / (5*Gamma[1/4])], {n, 1, 15}] (* _Vaclav Kotesovec_, Apr 16 2018 *)
%o A302914 (PARI) a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(4*i-3)))); \\ _Michel Marcus_, Apr 16 2018
%Y A302914 Cf. A001107.
%Y A302914 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).
%Y A302914 Cf. A007840 (permanent instead of determinant, for k=2).
%K A302914 nonn
%O A302914 1,2
%A A302914 _Muniru A Asiru_, Apr 15 2018