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A202768 Vandermonde determinant of the first n squares.

%I A202768 #43 Nov 27 2024 11:34:42
%S A202768 1,1,3,120,151200,10973491200,73004442255360000,
%T A202768 64942882916646518784000000,10615517921765466641283416064000000000,
%U A202768 419534029722194863260820186269027926016000000000000,5103425917047830280023316797736216735574814664897331200000000000000
%N A202768 Vandermonde determinant of the first n squares.
%C A202768 Each term divides its successor, as in A110468.
%C A202768 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
%H A202768 Alois P. Heinz, <a href="/A202768/b202768.txt">Table of n, a(n) for n = 0..29</a>
%F A202768 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
%F A202768 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
%F A202768 a(n) = A168467(n) / A000142(n). - _Federico Provvedi_, Apr 01 2021
%F A202768 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
%e A202768 a(3) = (4-1)(9-1)(9-4) = 120.
%p A202768 with(LinearAlgebra):
%p A202768 a:= n-> Determinant(VandermondeMatrix([i^2$i=1..n])):
%p A202768 seq(a(n), n=0..12);  # _Alois P. Heinz_, Aug 21 2014
%t A202768 f[j_] := j^2; z = 15;
%t A202768 v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
%t A202768 Table[v[n], {n, 1, z}]    (* A202768 *)
%t A202768 Table[v[n + 1]/v[n], {n, 1, z - 1}] (* A110468 *)
%t A202768 (* or *)
%t A202768 Det@TrigExpand@Array[#1^(2*#2)*Cosh[2*#2*ArcCsch[#1]]&,{#,#},{1,0}]&/@Range@16 (* _Federico Provvedi_, Jan 20 2021 *)
%t A202768 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 *)
%o A202768 (PARI) a(n)=prod(k=1,n,(2*k-1)!/k) /* _Paul D. Hanna_, Jan 02 2012 */
%o A202768 (Python)
%o A202768 from math import prod
%o A202768 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)<<n-1)//n if n else 1 # _Chai Wah Wu_, Nov 26 2023
%Y A202768 Cf. A000290, A110468, A168467, A000142.
%K A202768 nonn
%O A202768 0,3
%A A202768 _Clark Kimberling_, Jan 01 2012
%E A202768 a(0) from _Alois P. Heinz_, Aug 21 2014