A203425 -id:A203425 - cp's OEIS frontend

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

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

Offset: 1

Clark Kimberling, Jan 02 2012

Each term divides its successor, as in A203425.

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

Cf. A203425, A203421.

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

(* 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 *)

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

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

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

A203427 a(n) = w(n+1)/(4*w(n)), where w = A203426.

Original entry on oeis.org

-3, 48, -1000, 25920, -806736, 29360128, -1224440064, 57600000000, -3018173044480, 174359297654784, -11011033460963328, 754709361539940352, -55801305000000000000, 4427218577690292387840, -375183514207494575620096, 33824309717272203758665728, -3232463698006063164519284736, 326417514496000000000000000000

Offset: 1

Clark Kimberling, Jan 02 2012

sign

G. C. Greubel, Table of n, a(n) for n = 1..345

Cf. A203425, A203426.

[(-2*(n+2))^n*(n+1)/4: n in [1..20]]; // G. C. Greubel, Dec 05 2023

(* First program *)
f[j_]:= 1/(2 j + 2); z = 12;
v[n_]:= Product[Product[f[k] - f[j], {j, k-1}], {k, 2, n}];
1/Table[v[n], {n, z}]               (* A203426 *)
Table[v[n]/(4 v[n + 1]), {n, z}]    (* A203427 *)
(* Second program *)
Table[(-2*(n+2))^n*(n+1)/4, {n,20}] (* G. C. Greubel, Dec 05 2023 *)

[(-2*(n+2))^n*(n+1)/4 for n in range(1,21)] # G. C. Greubel, Dec 05 2023

a(n) = (1/4) * (n+1) * (-2*(n+2))^n. - Andrei Asinowski, Nov 03 2015

Name corrected by Andrei Asinowski, Nov 03 2015

Terms a(14) onward added by G. C. Greubel, Dec 05 2023

cp's OEIS Frontend

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

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