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A203426 Reciprocal of Vandermonde determinant of (1/4,1/6,...,1/(2n+2)).

%I A203426 #26 Dec 05 2023 08:34:44
%S A203426 1,-12,-2304,9216000,955514880000,-3083393008926720000,
%T A203426 -362115253665574567280640000,1773553697494609431031516590243840000,
%U A203426 408626771902758012909661422392180736000000000000,-4933225232839126697329071833709661506078108549120000000000000
%N A203426 Reciprocal of Vandermonde determinant of (1/4,1/6,...,1/(2n+2)).
%C A203426 Each term divides its successor, as in A203427.
%H A203426 G. C. Greubel, <a href="/A203426/b203426.txt">Table of n, a(n) for n = 1..34</a>
%F A203426 a(n) = Product_{k=1..n} k * (-2(k+1))^(k-1). - _Andrei Asinowski_, Nov 03 2015
%F A203426 a(n) ~ (-1)^(n*(n-1)/2) * A * 2^(n^2/2 - n/2 - 1/2) * n^(n^2/2 + n/2 - 17/12) / (sqrt(Pi) * exp(n^2/4 - n - 1)), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Dec 05 2015
%F A203426 a(n) = (-2)^binomial(n,2) * n! * (Gamma(n+2))^n / BarnesG(n+3). - _G. C. Greubel_, Dec 05 2023
%p A203426 with(LinearAlgebra):
%p A203426 a:= n-> 1/Determinant(VandermondeMatrix([1/(2*i+2)$i=1..n])):
%p A203426 seq(a(n), n=1..12);  # _Alois P. Heinz_, Jul 23 2017
%t A203426 (* First program *)
%t A203426 f[j_] := 1/(2 j + 2); z = 12;
%t A203426 v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}];
%t A203426 1/Table[v[n], {n, 1, z}]              (* A203426 *)
%t A203426 Table[v[n]/(4 v[n + 1]), {n, 1, z}]   (* A203427 *)
%t A203426 (* Second program *)
%t A203426 Table[(-2)^Binomial[n,2]*n!*(Gamma[n+2])^n/BarnesG[n+3], {n,20}] (* _G. C. Greubel_, Dec 05 2023 *)
%o A203426 (Magma)
%o A203426 BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
%o A203426 A203426:= func< n | (-2)^Binomial(n,2)*Factorial(n)*(Factorial(n+1))^n/BarnesG(n+3) >;
%o A203426 [A203426(n): n in [1..20]]; // _G. C. Greubel_, Dec 05 2023
%o A203426 (SageMath)
%o A203426 def BarnesG(n): return product(factorial(k) for k in range(n-1))
%o A203426 def A203426(n): return (-2)^binomial(n,2)*gamma(n+1)*(gamma(n+2))^n/BarnesG(n+3)
%o A203426 [A203426(n) for n in range(1,21)] # _G. C. Greubel_, Dec 05 2023
%Y A203426 Cf. A203421, A203424, A203427.
%K A203426 sign
%O A203426 1,2
%A A203426 _Clark Kimberling_, Jan 02 2012