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

%I #28 Dec 07 2023 08:27:52

%S 1,-4,-144,73728,737280000,-183458856960000,-1381360067999170560000,

%T 370806019753548356895375360000,

%U 4086267719027580129096614223921807360000,-2092169072142121026097466482647965368320000000000000

%N Reciprocal of Vandermonde determinant of (1/2,1/4,...,1/(2n)).

%C Each term divides its successor, as in A203425.

%H G. C. Greubel, <a href="/A203424/b203424.txt">Table of n, a(n) for n = 1..35</a>

%F a(n) = Product_{k=1..n} (-2k)^(k-1). - _Andrei Asinowski_, Nov 03 2015

%F a(n) ~ (-1)^(n*(n-1)/2) * A * 2^(n^2/2 - n/2 - 1/2) * n^(n^2/2 - n/2 - 5/12) / (sqrt(Pi) * exp(n^2/4-n)), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Dec 05 2015

%F a(n) = 2^binomial(n,2) * A203421(n). - _Kevin Ryde_, May 03 2022

%F a(n) = (-2)^binomial(n,2) * (n!)^n / BarnesG(n+2). - _G. C. Greubel_, Dec 07 2023

%t (* First program *)

%t f[j_] := 1/(2 j); z = 16;

%t v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}];

%t 1/Table[v[n], {n, z}] (* A203424 *)

%t Table[v[n]/(4 v[n + 1]), {n, z}] (* A203425 *)

%t (* Second program *)

%t Table[(-2)^Binomial[n,2]*(n!)^n/BarnesG[n+2], {n,20}] (* _G. C. Greubel_, Dec 07 2023 *)

%o (PARI) a(n) = prod(k=2,n, (-k)^(k-1)) << binomial(n,2); \\ _Kevin Ryde_, May 03 2022

%o (Magma)

%o BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;

%o A203424:= func< n| (-2)^Binomial(n, 2)*(Factorial(n))^n/BarnesG(n+2) >;

%o [A203424(n): n in [1..20]]; // _G. C. Greubel_, Dec 07 2023

%o (SageMath)

%o def BarnesG(n): return product(factorial(k) for k in range(n-1))

%o def A203424(n): return (-2)^binomial(n, 2)*(gamma(n+1))^n/BarnesG(n+2)

%o [A203424(n) for n in range(1, 21)] # _G. C. Greubel_, Dec 07 2023

%Y Cf. A203425, A203421.

%K sign,easy

%O 1,2

%A _Clark Kimberling_, Jan 02 2012