The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A152665 Number of leading even entries in all permutations of {1,2,...,n}. 2
 0, 1, 2, 16, 60, 540, 3024, 32256, 241920, 3024000, 28512000, 410572800, 4670265600, 76281004800, 1017080064000, 18598035456000, 284549942476800, 5762136335155200, 99527809425408000, 2211729098342400000, 42575785143091200000, 1030334000462807040000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Table of n, a(n) for n=1..22. 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 EXAMPLE 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. 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 *) CROSSREFS Cf. A152662, A152663, A152664. Sequence in context: A207688 A208495 A207583 * A183762 A061608 A212899 Adjacent sequences: A152662 A152663 A152664 * A152666 A152667 A152668 KEYWORD nonn AUTHOR Emeric Deutsch, Dec 13 2008 EXTENSIONS Examples expanded by N. J. A. Sloane, Sep 09 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 19 14:45 EDT 2024. Contains 372698 sequences. (Running on oeis4.)