A364849 Number of even parity alternating permutations of [1..n] starting with an odd integer.
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
Keywords
Examples
For n=5, a(5)=6 because we have 12345, 14523, 34125, 32541, 52143, 54321.
Links
- F. G. Kebede and F. Rakotondrajao, Parity Alternating Permutations Starting With an Odd Integer, Enumerative Combinatorics and Applications, 1:2 (2021) Article #S2R16.
Crossrefs
Cf. A010551.
Programs
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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
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.
Comments