OFFSET
1,3
COMMENTS
a(n) = Sum(k*A134434(n,k), k=0..floor(n/2)).
The average of the number of even entries that start a descent over all permutations of {1,2,...n} is (1/n)[floor(n/2)]^2.
REFERENCES
S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.
FORMULA
a(2n) = n(2n)!/2; a(2n+1) = n^2*(2n)!.
D-finite with recurrence (-4*n+11)*a(n) +(9*n-25)*a(n-1) +(n-2)*(4*n^2-3*n-3)*a(n-2) -(n-2)*(n-3)*(5*n-7)*a(n-3)=0. - R. J. Mathar, Jul 31 2022
EXAMPLE
a(3)=2 because the permutations of {1,2,3} are 123, 132, 2'13, 231, 312 and 32'1 with the even entries that start a descent marked.
MAPLE
a:=proc(n) if `mod`(n, 2)=0 then (1/4)*n*factorial(n) else (1/4)*(n-1)^2*factorial(n-1) end if end proc: seq(a(n), n=1..20);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Nov 16 2008
STATUS
approved