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A086205 Determinant of n X n matrix M_(i,j)=binomial(i^2, j). 5

%I

%S 1,1,6,360,302400,4572288000,1520925880320000,13153940405570764800000,

%T 3412910854477066178396160000000,

%U 30107378079113824305786648526848000000000

%N Determinant of n X n matrix M_(i,j)=binomial(i^2, j).

%C Closed form can be deduced from the conjecture that the count of triangular semi-standard Young tableaux with shape (n,..,1) and max part n equals 2^((n-1)n/2); see Mathematica line. - _Wouter Meeussen_, Nov 26 2017

%F a(n) = Product_{k=1..n} (2*k-1)!/(k-1)!.

%F a(n) ~ A^(1/2) * 2^(n^2 + n/2 + 5/24) * n^(n^2/2 + n/2 + 1/24) / exp(3*n^2/4 + n/2 + 1/24), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Jul 11 2015

%t Table[Product[(2*k-1)!/(k-1)!,{k,1,n}],{n,0,10}] (* _Vaclav Kotesovec_, Jul 11 2015 *)

%t Table[Product[ -i+j+n, {i,n}, {j, 1-i+n}], {n,0,10}];

%t Round[Table[Sqrt[Glaisher]/(2^(1/24 - n^2)* E^(1/24 + Derivative[1, 0][Zeta][-1, 1/2 + n])* Pi^(1/4 + n/2)*Gamma[1/2 + n]^(-(1/2) - n)), {n, 16}]] (* see comments *) (* _Wouter Meeussen_, Nov 26 2017 *)

%t Table[Glaisher^(3/2) * 2^(n^2 - 1/24) * BarnesG[n + 3/2] / (E^(1/8) * Pi^(n/2 + 1/4)), {n, 0, 12}] (* _Vaclav Kotesovec_, Mar 24 2019 *)

%o (PARI) a(n)=prod(k=1,n,(2*k-1)!/(k-1)!)

%o (PARI) a(n)=matdet(matrix(n,n,i,j,binomial(i^2,j)))

%K nonn

%O 0,3

%A _Benoit Cloitre_, Aug 27 2003

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Last modified October 20 03:04 EDT 2021. Contains 348099 sequences. (Running on oeis4.)