A202768 Vandermonde determinant of the first n squares.

1, 1, 3, 120, 151200, 10973491200, 73004442255360000, 64942882916646518784000000, 10615517921765466641283416064000000000, 419534029722194863260820186269027926016000000000000, 5103425917047830280023316797736216735574814664897331200000000000000

Offset: 0

Author

Clark Kimberling, Jan 01 2012

Keywords

• nonn

Comments

Each term divides its successor, as in A110468.

a(m) is also the determinant of m X m matrix M(i,j) = i^(2*j)*cosh(2*j*arccsch(i)), with i from 1 to m, and j from 0 to m-1. - Federico Provvedi, Jan 20 2021

Examples

			a(3) = (4-1)(9-1)(9-4) = 120.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..29

Crossrefs

Cf. A000290, A110468, A168467, A000142.

Programs

Maple

with(LinearAlgebra):
a:= n-> Determinant(VandermondeMatrix([i^2$i=1..n])):
seq(a(n), n=0..12);  # Alois P. Heinz, Aug 21 2014

Mathematica

f[j_] := j^2; z = 15;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]    (* A202768 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}] (* A110468 *)
(* or *)
Det@TrigExpand@Array[#1^(2*#2)*Cosh[2*#2*ArcCsch[#1]]&,{#,#},{1,0}]&/@Range@16 (* Federico Provvedi, Jan 20 2021 *)
Table[Exp[(n^2-1/24)*Log[2]-(n/2+1/4)*Log[Pi]+3/2*Log@Glaisher+Log@BarnesG[1+n]+Log@BarnesG[3/2+n]-1/8]/n!,{n, 0, 40}] (* Federico Provvedi, Apr 01 2021 after Vaclav Kotesovec's formula *)

PARI

a(n)=prod(k=1,n,(2*k-1)!/k) /* Paul D. Hanna, Jan 02 2012 */

Python

from math import prod
def A202768(n): return (prod(((m:=k+1<<1)*(m+1))**(n-1-k)//(k+1) for k in range(1,n-1))*3**(n-1)<Chai Wah Wu, Nov 26 2023

Formula

a(n) = Product_{k=0..n-1} (2*k+1)!/(k+1) = Product_{k=0..n-1} A110468(k). - Paul D. Hanna, Jan 02 2012

a(n) ~ 2^(n^2 + n - 7/24) * n^(n^2 - n/2 - 13/24) * Pi^((n-1)/2) / (sqrt(A) * exp(3*n^2/2 - n/2 - 1/24)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 25 2019

a(n) = A168467(n) / A000142(n). - Federico Provvedi, Apr 01 2021

For n > 0, a(n) = sqrt(BarnesG(2*n)) * Gamma(2*n) / (n * Gamma(n)^(3/2) * 2^((n-1)/2)). - Vaclav Kotesovec, Nov 27 2024

Extensions

a(0) from Alois P. Heinz, Aug 21 2014