A067624 a(n) = 2^(2*n)*(2*n)!.

1, 8, 384, 46080, 10321920, 3715891200, 1961990553600, 1428329123020800, 1371195958099968000, 1678343852714360832000, 2551082656125828464640000, 4714400748520531002654720000, 10409396852733332453861621760000

Offset: 0

Author

Benoit Cloitre, Feb 02 2002

Keywords

• nonn

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

Crossrefs

Cf. A000165.

Appears in A162445, A061549 and A120738. - Johannes W. Meijer, Jul 06 2009

Programs

Magma

[2^(2*n)*Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, Feb 18 2018

Maple

for n from 0 to 30 by 2 do printf(`%d,`,2^(n)*(n)!) od: # James Sellers, Feb 11 2002
A067624 := n -> 2^(2*n)*(2*n)!: seq(A067624(n), n=0..12); # Johannes W. Meijer, Jan 05 2017

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 *)

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).

a(n) = (A280442(n)/A046161(n))/(A223067(n)/A223068(n)). - Johannes W. Meijer, Jan 05 2017

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)

Extensions

More terms from James Sellers, Feb 11 2002