A085530 - OEIS

%I #25 Sep 08 2022 08:45:11

%S 1,9,625,117649,43046721,25937424601,23298085122481,29192926025390625,

%T 48661191875666868481,104127350297911241532841,

%U 278218429446951548637196401,907846434775996175406740561329,3552713678800500929355621337890625,16423203268260658146231467800709255289

%N a(n) = (2n+1)^(2n).

%C a(n)/4^n is the square of the determinant of a (2*n+1) X (2*n+1) matrix with elements M(j,k) = cos(Pi*j*k/n). See the MathOverflow link. - _Hugo Pfoertner_, Sep 18 2021

%H Vincenzo Librandi, <a href="/A085530/b085530.txt">Table of n, a(n) for n = 0..190</a>

%H Zhi-Wei Sun, Fedor Petrov, <a href="https://mathoverflow.net/questions/321098/">A surprising identity</a>, discussion in MathOverflow, Jan 17 2019.

%F From _Mathew Englander_, Aug 14 2020: (Start)

%F a(n) = A085527(n)^2.

%F a(n) = A085529(n)/(2*n + 1).

%F (End)

%F From _Alois P. Heinz_, Aug 14 2020: (Start)

%F a(n) = A016754(n)^n.

%F a(n) = A005408(n)^A005843(n). (End)

%t Table[(2 n + 1)^(2 n), {n, 0, 20}] (* _Vincenzo Librandi_, Feb 26 2013 *)

%o (Magma) [(2*n+1)^(2*n): n in [0..13]]; // _Vincenzo Librandi_, Feb 26 2013

%Y Cf. A005408, A005843, A016754, A085527, A085529.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jul 05 2003