A203424 Reciprocal of Vandermonde determinant of (1/2,1/4,...,1/(2n)).

1, -4, -144, 73728, 737280000, -183458856960000, -1381360067999170560000, 370806019753548356895375360000, 4086267719027580129096614223921807360000, -2092169072142121026097466482647965368320000000000000

Offset: 1

Author

Clark Kimberling, Jan 02 2012

Keywords

• sign
• easy

Comments

Each term divides its successor, as in A203425.

Links

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

Crossrefs

Cf. A203425, A203421.

Programs

Magma

BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
A203424:= func< n| (-2)^Binomial(n, 2)*(Factorial(n))^n/BarnesG(n+2) >;
[A203424(n): n in [1..20]]; // G. C. Greubel, Dec 07 2023

Mathematica

(* First program *)
f[j_] := 1/(2 j); z = 16;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}];
1/Table[v[n], {n, z}]             (* A203424 *)
Table[v[n]/(4 v[n + 1]), {n, z}]  (* A203425 *)
(* Second program *)
Table[(-2)^Binomial[n,2]*(n!)^n/BarnesG[n+2], {n,20}] (* G. C. Greubel, Dec 07 2023 *)

PARI

a(n) = prod(k=2,n, (-k)^(k-1)) << binomial(n,2); \\ Kevin Ryde, May 03 2022

SageMath

def BarnesG(n): return product(factorial(k) for k in range(n-1))
def A203424(n): return (-2)^binomial(n, 2)*(gamma(n+1))^n/BarnesG(n+2)
[A203424(n) for n in range(1, 21)] # G. C. Greubel, Dec 07 2023

Formula

a(n) = Product_{k=1..n} (-2k)^(k-1). - Andrei Asinowski, Nov 03 2015

a(n) ~ (-1)^(n*(n-1)/2) * A * 2^(n^2/2 - n/2 - 1/2) * n^(n^2/2 - n/2 - 5/12) / (sqrt(Pi) * exp(n^2/4-n)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Dec 05 2015

a(n) = 2^binomial(n,2) * A203421(n). - Kevin Ryde, May 03 2022

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