A048099 Number of degree-n even permutations of order exactly 2.

0, 0, 0, 3, 15, 45, 105, 315, 1323, 5355, 18315, 63855, 272415, 1264263, 5409495, 22302735, 101343375, 507711375, 2495918223, 11798364735, 58074029055, 309240315615, 1670570920095, 8792390355903, 46886941456575, 264381946998975, 1533013006902975, 8785301059346175, 50439885753378303

Offset

1,4

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Koda, Tatsuhiko; Sato, Masaki; Takegahara, Yugen; 2-adic properties for the numbers of involutions in the alternating groups, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages).

Formula

a(n) = (A001189(n) + A051684(n))/2.

a(n) = Sum_{i=1..floor(n/4)} binomial(n,4i)(4i)!/(2^(2i)(2i)!). - Luis Manuel Rivera Martínez, May 16 2018

E.g.f.: (exp(x + x^2/2) + exp(x - x^2/2))/2 - exp(x). - Andrew Howroyd, Feb 01 2020

Mathematica

Table[Sum[Binomial[n , 4 i] (4 i)!/(2^(2 i) (2 i)!), {i, 1, Floor[n/4]}], {n, 1, 22}] (* Luis Manuel Rivera Martínez, May 16 2018 *)

Prog

(PARI) a(n) = sum(i=1, n\4, binomial(n, 4*i)*(4*i)!/(2^(2*i)*(2*i)!)); \\ Michel Marcus, May 17 2018
(PARI) seq(n)={my(A=O(x*x^n)); Vec(serlaplace(exp(x + x^2/2 + A) + exp(x - x^2/2 + A) - 2*exp(x + A))/2, -n)} \\ Andrew Howroyd, Feb 01 2020

Crossrefs

Cf. A001189, A051695. A column of A057740.

Sequence in context: A334078 A094191 A050534 * A030505 A301632 A074355

Adjacent sequences: A048096 A048097 A048098 * A048100 A048101 A048102

Keyword

easy,nonn

Author

Vladeta Jovovic

Status

approved