A203422 Reciprocal of Vandermonde determinant of (1/2,1/3,...,1/(n+1)).
1, -6, -288, 144000, 933120000, -94097687040000, -172670008499896320000, 6607002383077924814192640000, 5946302144770132332773376000000000000, -140210694122490812598274255654748160000000000000
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..35
Programs
-
Magma
BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >; A203422:= func< n | (-1)^Binomial(n,2)*Factorial(n)*(Factorial(n+1))^n/BarnesG(n+3) >; [A203422(n): n in [1..20]]; // G. C. Greubel, Dec 08 2023
-
Mathematica
(* First program *) f[j_] := 1/(j + 1); z = 16; v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}] 1/Table[v[n], {n, z}] (* A203422 *) Table[v[n]/(2 v[n + 1]), {n, z}] (* A203423 *) (* Second program *) Table[(-1)^Binomial[n,2]*n!*(Gamma[n+2])^n/BarnesG[n+3], {n,20}] (* G. C. Greubel, Dec 08 2023 *) -
PARI
a(n) = my(f=n+1); prod(i=-n,-2, f*=i); \\ Kevin Ryde, Apr 17 2022
-
SageMath
def BarnesG(n): return product(factorial(k) for k in range(n-1)) def A203422(n): return (-1)^binomial(n,2)*gamma(n+1)*(gamma(n+2))^n/BarnesG(n+3) [A203422(n) for n in range(1, 21)] # G. C. Greubel, Dec 08 2023
Formula
a(n) = (n+1)^(n-1) * Product_{i=2..n} (-i)^(i-1). - Kevin Ryde, Apr 17 2022
a(n) = (-1)^binomial(n,2) * n! * (Gamma(n+2))^n / BarnesG(n+3). - G. C. Greubel, Dec 08 2023
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
Comments