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A048099 Number of degree-n even permutations of order exactly 2. 18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
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
KEYWORD
easy,nonn
AUTHOR
STATUS
approved

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Last modified March 28 11:44 EDT 2024. Contains 371241 sequences. (Running on oeis4.)