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A203428 Reciprocal of Vandermonde determinant of (1/3,1/6,...,1/(3n)).

%I A203428 #12 Aug 09 2025 05:48:33
%S A203428 1,-6,-486,839808,42515280000,-80335512599040000,
%T A203428 -6890065294166289123840000,31601087581187838970614157148160000,
%U A203428 8925080517850366815864624583251321642024960000
%N A203428 Reciprocal of Vandermonde determinant of (1/3,1/6,...,1/(3n)).
%C A203428 Each term divides its successor, as in A203429.
%H A203428 G. C. Greubel, <a href="/A203428/b203428.txt">Table of n, a(n) for n = 1..33</a>
%F A203428 a(n) = (-3)^binomial(n,2) * (Gamma(n+1))^(n-1) / BarnesG(n+1). - _G. C. Greubel_, Sep 28 2023
%F A203428 a(n) ~ (-1)^(n*(n-1)/2) * A * 3^(n*(n-1)/2) * n^(n*(n-1)/2 - 5/12) / (sqrt(2*Pi) * exp(n^2/4 - n)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Aug 09 2025
%t A203428 (* First program *)
%t A203428 f[j_]:= 1/(3*j); z = 16;
%t A203428 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A203428 1/Table[v[n], {n,z}]             (* A203428 *)
%t A203428 Table[v[n]/(3*v[n+1]), {n,z}]    (* A203429 *)
%t A203428 (* Second program *)
%t A203428 Table[(-3)^Binomial[n,2]*(Gamma[n+1])^(n-1)/BarnesG[n+1], {n,20}] (* _G. C. Greubel_, Sep 28 2023 *)
%o A203428 (Magma)
%o A203428 Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
%o A203428 A203428:= func< n | (-3)^Binomial(n,2)*(Factorial(n))^n/Barnes(n+1) >;
%o A203428 [A203428(n): n in [1..25]]; // _G. C. Greubel_, Sep 28 2023
%o A203428 (SageMath)
%o A203428 def barnes(n): return product(factorial(j) for j in range(n))
%o A203428 def A203428(n): return (-3)^binomial(n,2)*(factorial(n))^n/barnes(n+1)
%o A203428 [A203428(n) for n in range(1,21)] # _G. C. Greubel_, Sep 28 2023
%Y A203428 Cf. A203421, A203424, A203429.
%K A203428 sign
%O A203428 1,2
%A A203428 _Clark Kimberling_, Jan 02 2012