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
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).
Programs
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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 MartÃnez, 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
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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) = Sum_{i=1..floor(n/4)} binomial(n,4i)(4i)!/(2^(2i)(2i)!). - Luis Manuel Rivera MartÃnez, May 16 2018
E.g.f.: (exp(x + x^2/2) + exp(x - x^2/2))/2 - exp(x). - Andrew Howroyd, Feb 01 2020