OFFSET
0,2
COMMENTS
For n >= 1, a(n) equals the absolute value of the determinant of the 4n X 4n matrix with i's along the superdiagonal (where i is the imaginary unit), and 2, 3, 4, ... 4*n along the subdiagonal, and 0's everywhere else. (See Mathematica code below.) - John M. Campbell, Jun 04 2011
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = A000165(2*n) where A000165(k) are the double factorial numbers 2^k*k!=(2k)!!. - Corrected by Johannes W. Meijer, Jul 05 2009
a(n) = (4*n)!! = 2^(2*n)*(2*n)!. - Johannes W. Meijer, Jul 06 2009
sqrt((1+cos(x))/2) = Sum_{n>=0} (-1)^n * x^(2*n) / a(n).
From Amiram Eldar, Jul 12 2020: (Start)
Sum_{n>=0} 1/a(n) = cosh(1/2).
Sum_{n>=0} (-1)^n/a(n) = cos(1/2). (End)
MAPLE
for n from 0 to 30 by 2 do printf(`%d, `, 2^(n)*(n)!) od: # James A. Sellers, Feb 11 2002
MATHEMATICA
Table[Abs[Det[Array[KroneckerDelta[#1 + 1, #2]*I &, {4*n, 4*n}] + Array[KroneckerDelta[#1 - 1, #2]*#1 &, {4*n, 4*n}]]], {n, 1, 20}] (* John M. Campbell, Jun 04 2011 *)
Table[2^(2 n) (2 n)!, {n, 0, 30}] (* Vincenzo Librandi, Feb 18 2018 *)
PROG
(Magma) [2^(2*n)*Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, Feb 18 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Feb 02 2002
EXTENSIONS
More terms from James A. Sellers, Feb 11 2002
STATUS
approved