A280921 Degree of SO(n,C), the special orthogonal group, as an algebraic variety.

2, 8, 40, 384, 4768, 111616, 3433600, 196968448, 14994641408, 2112561610752, 397713919469568, 137785594909556736, 64120367727755108352, 54666180849611078369280, 62864933930402036994048000, 131959858152100309567348408320, 374913851106401853810511580364800, 1938349609799484523235647407112847360, 13603397258157549964912652571654029312000

Offset: 2

Author

Taylor Brysiewicz, Jan 10 2017

Keywords

• nonn

Examples

			For n = 4 we have a(4) = 2^3*det({6,1},{1,1}) = 2^3*(6-1) = 40.
		

Links

• M. Brandt, D. Bruce, T. Brysiewicz, R. Krone, E. Robeva, The degree of SO(n), arXiv:1701.03200 [math.AG], 2017

Crossrefs

Cf. A086645, A103328, A280922, A280923.

Programs

Mathematica

a[n_] := 2^(n-1) Det[Table[Binomial[2n-2i-2j, n-2i], {i, n/2}, {j, n/2}]];
Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Aug 12 2018 *)

PARI

a(n) = 2^(n-1)*matdet(matrix(n\2,n\2,i,j,binomial(2*n-2*i-2*j,n-2*i))); \\ Michel Marcus, Jan 14 2017

Formula

a(n) = 2^(n-1)*det(binomial(2n-2i-2j, n-2i))_{i,j=1..floor(n/2)}.

a(2*n+1) = A280922(n) * 2^(2*n).

Let M_n be the n X n matrix M_n(i, j) = binomial(2*i+2*j-2, 2*i-1) = A103328(i+j-1, i-1); then a(2*n+1) = 2^(2*n)*det(M_n).

Let M_n be the n X n matrix M_n(i,j) = binomial(2*i+2*j-4, 2*i-2) = A086645(i+j-2, i-1); then a(2*n) = 2^(2*n-1)*det(M_n).