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A203422 Reciprocal of Vandermonde determinant of (1/2,1/3,...,1/(n+1)).

%I A203422 #17 Aug 09 2025 05:40:25
%S A203422 1,-6,-288,144000,933120000,-94097687040000,-172670008499896320000,
%T A203422 6607002383077924814192640000,5946302144770132332773376000000000000,
%U A203422 -140210694122490812598274255654748160000000000000
%N A203422 Reciprocal of Vandermonde determinant of (1/2,1/3,...,1/(n+1)).
%C A203422 Each term divides its successor, as in A203423.
%H A203422 G. C. Greubel, <a href="/A203422/b203422.txt">Table of n, a(n) for n = 1..35</a>
%F A203422 a(n) = (n+1)^(n-1) * Product_{i=2..n} (-i)^(i-1). - _Kevin Ryde_, Apr 17 2022
%F A203422 a(n) = (-1)^binomial(n,2) * n! * (Gamma(n+2))^n / BarnesG(n+3). - _G. C. Greubel_, Dec 08 2023
%F A203422 a(n) ~ (-1)^(n*(n-1)/2) * A * n^(n*(n+1)/2 - 17/12) / (sqrt(2*Pi) * exp(n^2/4 - n - 1)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Aug 09 2025
%t A203422 (* First program *)
%t A203422 f[j_] := 1/(j + 1); z = 16;
%t A203422 v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
%t A203422 1/Table[v[n], {n, z}]             (* A203422 *)
%t A203422 Table[v[n]/(2 v[n + 1]), {n, z}]  (* A203423 *)
%t A203422 (* Second program *)
%t A203422 Table[(-1)^Binomial[n,2]*n!*(Gamma[n+2])^n/BarnesG[n+3], {n,20}] (* _G. C. Greubel_, Dec 08 2023 *)
%o A203422 (PARI) a(n) = my(f=n+1); prod(i=-n,-2, f*=i); \\ _Kevin Ryde_, Apr 17 2022
%o A203422 (Magma)
%o A203422 BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
%o A203422 A203422:= func< n | (-1)^Binomial(n,2)*Factorial(n)*(Factorial(n+1))^n/BarnesG(n+3) >;
%o A203422 [A203422(n): n in [1..20]]; // _G. C. Greubel_, Dec 08 2023
%o A203422 (SageMath)
%o A203422 def BarnesG(n): return product(factorial(k) for k in range(n-1))
%o A203422 def A203422(n): return (-1)^binomial(n,2)*gamma(n+1)*(gamma(n+2))^n/BarnesG(n+3)
%o A203422 [A203422(n) for n in range(1, 21)] # _G. C. Greubel_, Dec 08 2023
%Y A203422 Cf. A203421, A203423.
%K A203422 sign,easy
%O A203422 1,2
%A A203422 _Clark Kimberling_, Jan 02 2012