%I #15 Dec 08 2023 09:58:32
%S 1,-6,-288,144000,933120000,-94097687040000,-172670008499896320000,
%T 6607002383077924814192640000,5946302144770132332773376000000000000,
%U -140210694122490812598274255654748160000000000000
%N Reciprocal of Vandermonde determinant of (1/2,1/3,...,1/(n+1)).
%C Each term divides its successor, as in A203423.
%H G. C. Greubel, <a href="/A203422/b203422.txt">Table of n, a(n) for n = 1..35</a>
%F a(n) = (n+1)^(n-1) * Product_{i=2..n} (-i)^(i-1). - _Kevin Ryde_, Apr 17 2022
%F a(n) = (-1)^binomial(n,2) * n! * (Gamma(n+2))^n / BarnesG(n+3). - _G. C. Greubel_, Dec 08 2023
%t (* First program *)
%t f[j_] := 1/(j + 1); 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}] (* A203422 *)
%t Table[v[n]/(2 v[n + 1]), {n, z}] (* A203423 *)
%t (* Second program *)
%t Table[(-1)^Binomial[n,2]*n!*(Gamma[n+2])^n/BarnesG[n+3], {n,20}] (* _G. C. Greubel_, Dec 08 2023 *)
%o (PARI) a(n) = my(f=n+1); prod(i=-n,-2, f*=i); \\ _Kevin Ryde_, Apr 17 2022
%o (Magma)
%o BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
%o A203422:= func< n | (-1)^Binomial(n,2)*Factorial(n)*(Factorial(n+1))^n/BarnesG(n+3) >;
%o [A203422(n): n in [1..20]]; // _G. C. Greubel_, Dec 08 2023
%o (SageMath)
%o def BarnesG(n): return product(factorial(k) for k in range(n-1))
%o def A203422(n): return (-1)^binomial(n,2)*gamma(n+1)*(gamma(n+2))^n/BarnesG(n+3)
%o [A203422(n) for n in range(1, 21)] # _G. C. Greubel_, Dec 08 2023
%Y Cf. A203421, A203423.
%K sign,easy
%O 1,2
%A _Clark Kimberling_, Jan 02 2012