A203426 Reciprocal of Vandermonde determinant of (1/4,1/6,...,1/(2n+2)).
1, -12, -2304, 9216000, 955514880000, -3083393008926720000, -362115253665574567280640000, 1773553697494609431031516590243840000, 408626771902758012909661422392180736000000000000, -4933225232839126697329071833709661506078108549120000000000000
Offset: 1
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 1..34
Programs
-
Magma
BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >; A203426:= func< n | (-2)^Binomial(n,2)*Factorial(n)*(Factorial(n+1))^n/BarnesG(n+3) >; [A203426(n): n in [1..20]]; // G. C. Greubel, Dec 05 2023
-
Maple
with(LinearAlgebra): a:= n-> 1/Determinant(VandermondeMatrix([1/(2*i+2)$i=1..n])): seq(a(n), n=1..12); # Alois P. Heinz, Jul 23 2017
-
Mathematica
(* First program *) f[j_] := 1/(2 j + 2); z = 12; v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]; 1/Table[v[n], {n, 1, z}] (* A203426 *) Table[v[n]/(4 v[n + 1]), {n, 1, z}] (* A203427 *) (* Second program *) Table[(-2)^Binomial[n,2]*n!*(Gamma[n+2])^n/BarnesG[n+3], {n,20}] (* G. C. Greubel, Dec 05 2023 *) -
SageMath
def BarnesG(n): return product(factorial(k) for k in range(n-1)) def A203426(n): return (-2)^binomial(n,2)*gamma(n+1)*(gamma(n+2))^n/BarnesG(n+3) [A203426(n) for n in range(1,21)] # G. C. Greubel, Dec 05 2023
Formula
a(n) = Product_{k=1..n} k * (-2(k+1))^(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 - 17/12) / (sqrt(Pi) * exp(n^2/4 - n - 1)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Dec 05 2015
a(n) = (-2)^binomial(n,2) * n! * (Gamma(n+2))^n / BarnesG(n+3). - G. C. Greubel, Dec 05 2023
Comments