A203428 Reciprocal of Vandermonde determinant of (1/3,1/6,...,1/(3n)).

1, -6, -486, 839808, 42515280000, -80335512599040000, -6890065294166289123840000, 31601087581187838970614157148160000, 8925080517850366815864624583251321642024960000

Offset: 1

Author

Clark Kimberling, Jan 02 2012

Keywords

• sign

Comments

Each term divides its successor, as in A203429.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..33

Crossrefs

Cf. A203421, A203424, A203429.

Programs

Magma

Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
A203428:= func< n | (-3)^Binomial(n,2)*(Factorial(n))^n/Barnes(n+1) >;
[A203428(n): n in [1..25]]; // G. C. Greubel, Sep 28 2023

Mathematica

(* First program *)
f[j_]:= 1/(3*j); z = 16;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
1/Table[v[n], {n,z}]             (* A203428 *)
Table[v[n]/(3*v[n+1]), {n,z}]    (* A203429 *)
(* Second program *)
Table[(-3)^Binomial[n,2]*(Gamma[n+1])^(n-1)/BarnesG[n+1], {n,20}] (* G. C. Greubel, Sep 28 2023 *)

SageMath

def barnes(n): return product(factorial(j) for j in range(n))
def A203428(n): return (-3)^binomial(n,2)*(factorial(n))^n/barnes(n+1)
[A203428(n) for n in range(1,21)] # G. C. Greubel, Sep 28 2023

Formula

a(n) = (-3)^binomial(n,2) * (Gamma(n+1))^(n-1) / BarnesG(n+1). - G. C. Greubel, Sep 28 2023

a(n) ~ (-1)^(n*(n-1)/2) * A * 3^(n*(n-1)/2) * 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, Aug 09 2025