A152665 Number of leading even entries in all permutations of {1,2,...,n}.

0, 1, 2, 16, 60, 540, 3024, 32256, 241920, 3024000, 28512000, 410572800, 4670265600, 76281004800, 1017080064000, 18598035456000, 284549942476800, 5762136335155200, 99527809425408000, 2211729098342400000, 42575785143091200000, 1030334000462807040000

Offset: 1

Author

Emeric Deutsch, Dec 13 2008

Keywords

• nonn

Examples

			The permutation 4,6,2,1,5,3  begins with three even numbers, so would contribute 3 to a(6).
a(3)=2 because in the permutations 123, 132, 213, 231, 312, 321 we have 0+0+1+1+0+0 = 2 leading odd entries.
a(45) = 16: Here are the permutations of 1234, 24 in all:
1(234) total 6, no. of initial even terms  = 0
3(124) ditto
21(34) total 2, no. of initial even terms 1*2 = 2
23(14) ditto
24(13) total 2, no. of initial even terms 2 twice = 4
Subtotal from 2*** is 2+2+4 = 8
Subtotal from 4*** is also 2+2+4 = 8
Total a(4) = 8+8 = 16.
		

Crossrefs

Cf. A152662, A152663, A152664.

Programs

Maple

ao := proc (n) options operator, arrow; n*factorial(2*n+1)/(n+2) end proc: ae := proc (n) options operator, arrow; n*factorial(2*n)/(n+1) end proc: a := proc (n) if `mod`(n, 2) = 1 then ao((1/2)*n-1/2) else ae((1/2)*n) end if end proc; seq(a(n), n = 1 .. 20);

Mathematica

a[n_] := If[OddQ[n], (n-1)*n!/(n+3), n*n!/(n+2)];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Apr 29 2023 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} k*A152664(n,k).

a(2n+1) = n(2n+1)!/(n+2);

a(2n) = n(2n)!/(n+1).

D-finite with recurrence 2*(n+3)*a(n) +(-5*n-8)*a(n-1) +(-2*n^3-2*n^2-n-4)*a(n-2) +(n-2)*(3*n^2-3*n+2)*a(n-3) +(n-3)*(n-2)^2*a(n-4)=0. - R. J. Mathar, Jul 26 2022

Extensions

Examples expanded by N. J. A. Sloane, Sep 09 2019