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

2,1

Links

Table of n, a(n) for n=2..20.

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A086645, A103328, A280922, A280923.

Sequence in context: A180736 A111394 A140363 * A208962 A351109 A012597

Adjacent sequences: A280918 A280919 A280920 * A280922 A280923 A280924

Keyword

nonn

Author

Taylor Brysiewicz, Jan 10 2017

Status

approved