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

1,5

Comments

a(n) is also the number of odd parity alternating permutations of [1..n] starting with an odd integer for n>2.

Links

Table of n, a(n) for n=1..26.

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

Adjacent sequences: A364846 A364847 A364848 * A364850 A364851 A364852

Keyword

nonn

Author

Frether Getachew Kebede, Aug 10 2023

Status

approved