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 A086205 Determinant of n X n matrix M_(i,j)=binomial(i^2, j). 5
 1, 1, 6, 360, 302400, 4572288000, 1520925880320000, 13153940405570764800000, 3412910854477066178396160000000, 30107378079113824305786648526848000000000 (list; graph; refs; listen; history; text; internal format)
 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 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 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 Sequence in context: A262179 A064350 A069945 * A173608 A042759 A188954 Adjacent sequences:  A086202 A086203 A086204 * A086206 A086207 A086208 KEYWORD nonn AUTHOR Benoit Cloitre, Aug 27 2003 STATUS approved

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Last modified September 26 13:34 EDT 2021. Contains 347668 sequences. (Running on oeis4.)