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

Author

Clark Kimberling, Jan 02 2012

Keywords

• sign
• easy

Comments

Each term divides its successor, as in A203423.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..35

Crossrefs

Cf. A203421, A203423.

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