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
KEYWORD
nonn
AUTHOR
Frether Getachew Kebede, Aug 10 2023
STATUS
approved