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

1, 1, -2, -18, 1152, 720000, -5598720000, -658683809280000, 1381360067999170560000, 59463021447701323327733760000, -59463021447701323327733760000000000000, -1542317635347398938581016812202229760000000000000

Offset: 0

Author

Clark Kimberling, Jan 02 2012

Keywords

• sign
• easy

Comments

Each term divides its successor, as in A000169.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..35

Crossrefs

Cf. A000169, A002109, A057077, A203422, A203424.

Programs

Magma

BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
A203421:= func< n | (-1)^Binomial(n,2)*(Factorial(n))^n/BarnesG(n+2) >;
[A203421(n): n in [1..20]]; // G. C. Greubel, Dec 07 2023

Mathematica

(* First program *)
f[j_] := 1/j; z = 12;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]
1/%  (* A203421 *)
Table[v[n]/v[n + 1], {n, 1, z}]  (* A000169 signed *)
(* Additional programs *)
Table[(-1)^Floor[n/2]*Product[(k + 1)^k, {k, 0, n-1}], {n, 1, 10}] (* Vaclav Kotesovec, Oct 18 2015 *)
Table[(-1)^Binomial[n,2]*(n!)^n/BarnesG[n+2], {n, 20}] (* G. C. Greubel, Dec 07 2023 *)

PARI

a(n) = prod(i=2,n, (-i)^(i-1)); \\ Kevin Ryde, Apr 17 2022

SageMath

def BarnesG(n): return product(factorial(k) for k in range(n-1))
def A203421(n): return (-1)^binomial(n, 2)*(gamma(n+1))^n/BarnesG(n+2)
[A203421(n) for n in range(1, 21)] # G. C. Greubel, Dec 07 2023

Formula

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

a(n) = (-1)^floor(n/2) * hyperfactorial(n)/n! = A057077(n) * A002109(n)/n!. - Paul J. Harvey, Feb 08 2014

a(n) = Product_{i=2..n} (-i)^(i-1). - Kevin Ryde, Apr 17 2022

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

a(n) = (-1)^binomial(n,2) * (n!)^n / BarnesG(n+2). - G. C. Greubel, Dec 07 2023

Extensions

a(0)=1 prepended by Alois P. Heinz, Apr 13 2024