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A364849
Number of even parity alternating permutations of [1..n] starting with an odd integer.
0
1, 1, 1, 1, 2, 6, 18, 72, 288, 1440, 7200, 43200, 259200, 1814400, 12700800, 101606400, 812851200, 7315660800, 65840947200, 658409472000, 6584094720000, 72425041920000, 796675461120000, 9560105533440000, 114721266401280000, 1491376463216640000
OFFSET
1,5
COMMENTS
a(n) is also the number of odd parity alternating permutations of [1..n] starting with an odd integer for n>2.
LINKS
F. G. Kebede and F. Rakotondrajao, Parity Alternating Permutations Starting With an Odd Integer, Enumerative Combinatorics and Applications, 1:2 (2021) Article #S2R16.
FORMULA
E.g.f.: 1/(2-x) + arccos(1-x^2/2)/(2*(2-x)*sqrt(1-x^2/4)) + x^2/4 + x/2 + 1/2.
a(n) = A010551(n)/2, for n>2.
EXAMPLE
For n=5, a(5)=6 because we have 12345, 14523, 34125, 32541, 52143, 54321.
PROG
(PARI) seq(n) = my(A=O(x*x^n)); Vec(serlaplace(1/(2-x) + acos(1-x^2/2+A)/(2*(2-x)*sqrt(1-x^2/4+A)) + x^2/4 + x/2 +1/2)) \\ Andrew Howroyd, Aug 10 2023
CROSSREFS
Cf. A010551.
Sequence in context: A264036 A261994 A177472 * A194088 A022491 A004395
KEYWORD
nonn
AUTHOR
STATUS
approved