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A280921 Degree of SO(n,C), the special orthogonal group, as an algebraic variety. 2
2, 8, 40, 384, 4768, 111616, 3433600, 196968448, 14994641408, 2112561610752, 397713919469568, 137785594909556736, 64120367727755108352, 54666180849611078369280, 62864933930402036994048000, 131959858152100309567348408320, 374913851106401853810511580364800, 1938349609799484523235647407112847360, 13603397258157549964912652571654029312000 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
LINKS
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
Sequence in context: A180736 A111394 A140363 * A208962 A351109 A012597
KEYWORD
nonn
AUTHOR
Taylor Brysiewicz, Jan 10 2017
STATUS
approved

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Last modified March 1 22:23 EST 2024. Contains 370443 sequences. (Running on oeis4.)