A203426 Reciprocal of Vandermonde determinant of (1/4,1/6,...,1/(2n+2)).

1, -12, -2304, 9216000, 955514880000, -3083393008926720000, -362115253665574567280640000, 1773553697494609431031516590243840000, 408626771902758012909661422392180736000000000000, -4933225232839126697329071833709661506078108549120000000000000

Offset

1,2

Comments

Each term divides its successor, as in A203427.

Links

G. C. Greubel, Table of n, a(n) for n = 1..34

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A203421, A203424, A203427.

Sequence in context: A268421 A129206 A135398 * A268589 A228289 A306391

Adjacent sequences: A203423 A203424 A203425 * A203427 A203428 A203429

Keyword

sign

Author

Clark Kimberling, Jan 02 2012

Status

approved