A145889 - OEIS

%I #9 Jul 31 2022 06:32:46

%S 0,1,2,24,96,1080,6480,80640,645120,9072000,90720000,1437004800,

%T 17244057600,305124019200,4271736268800,83691159552000,

%U 1339058552832000,28810681675776000,518592270163968000,12164510040883200000

%N Number of even entries that are followed by a smaller entry in all permutations of {1,2,...,n}.

%C a(n) = Sum(k*A134434(n,k), k=0..floor(n/2)).

%C 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.

%D S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.

%F a(2n) = n(2n)!/2; a(2n+1) = n^2*(2n)!.

%F 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

%e 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.

%p 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);

%Y Cf. A134434, A145890.

%K nonn,easy

%O 1,3

%A _Emeric Deutsch_, Nov 16 2008