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A203421 Reciprocal of Vandermonde determinant of (1,1/2,...,1/n).

%I A203421 #39 Apr 27 2024 09:30:36
%S A203421 1,1,-2,-18,1152,720000,-5598720000,-658683809280000,
%T A203421 1381360067999170560000,59463021447701323327733760000,
%U A203421 -59463021447701323327733760000000000000,-1542317635347398938581016812202229760000000000000
%N A203421 Reciprocal of Vandermonde determinant of (1,1/2,...,1/n).
%C A203421 Each term divides its successor, as in A000169.
%H A203421 G. C. Greubel, <a href="/A203421/b203421.txt">Table of n, a(n) for n = 0..35</a>
%F A203421 G.f.: G(0)/(2*x) -1/x, where G(k)= 1 + 1/(1 - x/(x + (2*k+1)/((2*k+1)^(2*k+1))/(1 + 1/(1 - x/(x - (2*k+2)/((2*k+2)^(2*k+2))/G(k+1)))))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 03 2013
%F A203421 a(n) = (-1)^floor(n/2) * hyperfactorial(n)/n! = A057077(n) * A002109(n)/n!. - _Paul J. Harvey_, Feb 08 2014
%F A203421 a(n) = Product_{i=2..n} (-i)^(i-1). - _Kevin Ryde_, Apr 17 2022
%F A203421 abs(a(n)) ~ A * 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_, Nov 20 2023
%F A203421 a(n) = (-1)^binomial(n,2) * (n!)^n / BarnesG(n+2). - _G. C. Greubel_, Dec 07 2023
%t A203421 (* First program *)
%t A203421 f[j_] := 1/j; z = 12;
%t A203421 v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
%t A203421 Table[v[n], {n, 1, z}]
%t A203421 1/%  (* A203421 *)
%t A203421 Table[v[n]/v[n + 1], {n, 1, z}]  (* A000169 signed *)
%t A203421 (* Additional programs *)
%t A203421 Table[(-1)^Floor[n/2]*Product[(k + 1)^k, {k, 0, n-1}], {n, 1, 10}] (* _Vaclav Kotesovec_, Oct 18 2015 *)
%t A203421 Table[(-1)^Binomial[n,2]*(n!)^n/BarnesG[n+2], {n, 20}] (* _G. C. Greubel_, Dec 07 2023 *)
%o A203421 (PARI) a(n) = prod(i=2,n, (-i)^(i-1)); \\ _Kevin Ryde_, Apr 17 2022
%o A203421 (Magma)
%o A203421 BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
%o A203421 A203421:= func< n | (-1)^Binomial(n,2)*(Factorial(n))^n/BarnesG(n+2) >;
%o A203421 [A203421(n): n in [1..20]]; // _G. C. Greubel_, Dec 07 2023
%o A203421 (SageMath)
%o A203421 def BarnesG(n): return product(factorial(k) for k in range(n-1))
%o A203421 def A203421(n): return (-1)^binomial(n, 2)*(gamma(n+1))^n/BarnesG(n+2)
%o A203421 [A203421(n) for n in range(1, 21)] # _G. C. Greubel_, Dec 07 2023
%Y A203421 Cf. A000169, A002109, A057077, A203422, A203424.
%K A203421 sign,easy
%O A203421 0,3
%A A203421 _Clark Kimberling_, Jan 02 2012
%E A203421 a(0)=1 prepended by _Alois P. Heinz_, Apr 13 2024