A301898 a(n) = (2*n + 1)! if n is even, a(n) = 2*(2*n + 1)! if n is odd.

12, 120, 10080, 362880, 79833600, 6227020800, 2615348736000, 355687428096000, 243290200817664000, 51090942171709440000, 51704033477769953280000, 15511210043330985984000000, 21777738900836704321536000000, 8841761993739701954543616000000, 16445677308355845635451125760000000

Offset

1,1

Comments

a(n) is the order of the first unstable homotopy group of Sp(n), namely pi_{4n+2}(Sp(n)), which is always finite cyclic.

This sequence can be produced by first forming the sequence consisting of the terms of A000142 corresponding to n > 1 odd. Reindex this new sequence starting at n = 1. Now double the terms of this sequence with n odd.

Links

Table of n, a(n) for n=1..15.

MathOverflow, The first unstable homotopy group of Sp(n).

Mamoru Mimura and Hiroshi Toda, Homotopy groups of symplectic groups, Journal of Mathematics of Kyoto University, Vol. 3, No. 2 (1963), 251-273.

Formula

a(n) = (2n+1)!*(3 + (-1)^(n+1))/2.

Sum_{n>=1} 1/a(n) = (sin(1) + 3*sinh(1))/4 - 1. - Amiram Eldar, Jun 30 2022

Example

For n = 2, we have a(2) = 120 as pi_{10}(Sp(2)) = Z_{120}.

Mathematica

a[n_] := If[EvenQ[n], (2*n + 1)!, 2*(2*n + 1)!]; Array[a, 15] (* Amiram Eldar, Jun 30 2022 *)

Prog

(PARI) a(n) = (2*n+1)!*(3 + (-1)^(n+1))/2 \\ Felix Fröhlich, Mar 28 2018

Crossrefs

Cf. A000142.

Sequence in context: A012436 A012619 A012563 * A012441 A012373 A012620

Adjacent sequences: A301895 A301896 A301897 * A301899 A301900 A301901

Keyword

easy,nonn

Author

Michael Albanese, Mar 28 2018

Status

approved