A048099 - OEIS

%I #30 Feb 01 2020 18:37:11

%S 0,0,0,3,15,45,105,315,1323,5355,18315,63855,272415,1264263,5409495,

%T 22302735,101343375,507711375,2495918223,11798364735,58074029055,

%U 309240315615,1670570920095,8792390355903,46886941456575,264381946998975,1533013006902975,8785301059346175,50439885753378303

%N Number of degree-n even permutations of order exactly 2.

%H Andrew Howroyd, <a href="/A048099/b048099.txt">Table of n, a(n) for n = 1..200</a>

%H Koda, Tatsuhiko; Sato, Masaki; Takegahara, Yugen; <a href="http://dx.doi.org/10.1142/S0219498815500528">2-adic properties for the numbers of involutions in the alternating groups</a>, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages).

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

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

%F E.g.f.: (exp(x + x^2/2) + exp(x - x^2/2))/2 - exp(x). - _Andrew Howroyd_, Feb 01 2020

%t 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 *)

%o (PARI) a(n) = sum(i=1, n\4, binomial(n,4*i)*(4*i)!/(2^(2*i)*(2*i)!)); \\ _Michel Marcus_, May 17 2018

%o (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

%Y Cf. A001189, A051695. A column of A057740.

%K easy,nonn

%O 1,4

%A _Vladeta Jovovic_