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

1, 1, 6, 360, 302400, 4572288000, 1520925880320000, 13153940405570764800000, 3412910854477066178396160000000, 30107378079113824305786648526848000000000

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..9.

Formula

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

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

a(n) = s_lambda(1,2,3,...,n) where s is the Schur polynomial in n variables and lambda is the partition (n,n-1,n-2,...,1). - Leonid Bedratyuk, Feb 06 2022

From Peter Bala, Feb 19 2023: (Start)

a(n) = Product_{k = 0..n} (n + k)!/(2*k)!. Cf. A266091.

a(n) = 2^(n*(n-1)/2) * A057863(n).

a(n+1)*a(n-1) = (4*n + 2)*a(n)^2.

a(n)^3*a(n+2) + a(n-1)*a(n+1)^3 = (8*n + 8)*(a(n)*a(n + 1))^2 for n >= 1.

Conjecture: a(n) = the determinant of the n X n matrix ( binomial(i^2 + z, j) ) for arbitrary complex z. (End)

Mathematica

Table[Product[(2*k-1)!/(k-1)!, {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Jul 11 2015 *)
Table[Product[ -i+j+n, {i, n}, {j, 1-i+n}], {n, 0, 10}];
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 *)
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 *)

Prog

(PARI) a(n)=prod(k=1, n, (2*k-1)!/(k-1)!)
(PARI) a(n)=matdet(matrix(n, n, i, j, binomial(i^2, j)))

Crossrefs

Cf. A057863, A266091.

Sequence in context: A262179 A064350 A069945 * A173608 A042759 A188954

Adjacent sequences: A086202 A086203 A086204 * A086206 A086207 A086208

Keyword

nonn,easy

Author

Benoit Cloitre, Aug 27 2003

Status

approved