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

Author

Vladeta Jovovic

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A001189, A051695. A column of A057740.

Programs

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 *)

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

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