A052582 a(n) = 2*n*n!.

0, 2, 8, 36, 192, 1200, 8640, 70560, 645120, 6531840, 72576000, 878169600, 11496038400, 161902540800, 2440992153600, 39230231040000, 669529276416000, 12093372555264000, 230485453406208000, 4622513815535616000, 97316080327065600000, 2145819571211796480000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Total number of pairs (a_i,a_(i+1)) in all permutations on [n] such that a_i,a_(i+1) are consecutive integers. - David Callan, Nov 04 2003

Number of permutations of {1,2,...,n+2} such that there is exactly one entry between the entries 1 and 2. Example: a(2)=8 because we have 1324, 1423, 2314, 2413, 3142, 4132, 3241 and 4231. - Emeric Deutsch, Apr 06 2008

Number of permutations of 0 to n distinct letters (ABC...) 1 times ("-" (0), A (1), AB (1-1), ABC (1-1-1), ABCD (1-1-1-1 )etc...) and one after the other to resemble motif:( "-",... BB (0-2), ABB (1-2-0), AABB (2-2-0-0), AAABB (3-2-0-0-0) AAAABB (4-2-0-0-0-0), AAAAABB (5-2-0-0-0-0-0), AAAAAABB (6-2-0-0-0-0-0-0), etc... 0 fixed point (or free fixed point). Example: if ABC (1-1-1) and motif ABB (1-2-0) then 2 * 0 (free) fixed point, if ABCD (1-1-1-1), and motif AABB (2-2-0-0) then 8 * 0 (free) fixed point, if ABCDE (1-1-1-1-1), and motif AAABB (3-2-0-0-0), then 36 * 0 (free) fixed point, if ABCDEF (1-1-1-1-1-1), and motif AAAABB (4-2-0-0-0-0), then 192 * 0 (free) fixed point, if ABCDEFG (1-1-1-1-1-1-1), and motif AAAAABB (5-2-0-0-0-0-0), then 1200 * 0 (free) fixed point, etc... - Zerinvary Lajos, Dec 07 2009

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..250
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 526.
• Eric Weisstein's World of Mathematics, Exponential Integral.

Crossrefs

Cf. A001339, A001563, A001620, A007808, A138770, A000142, A000290, A091725, A099285.

Programs

Haskell

a052582 n = a052582_list !! n
a052582_list =  0 : 2 : zipWith
   div (zipWith (*) (tail a052582_list) (drop 2 a000290_list)) [1..]
-- Reinhard Zumkeller, Nov 12 2011

Maple

spec := [S,{S=Prod(Sequence(Z),Sequence(Z),Union(Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

a[ n_] := If[ n<0, 0, n! SeriesCoefficient[ 2 x / (1 - x)^2, {x, 0, n}]]; (* Michael Somos, Oct 20 2011 *)
a[ n_] := If[ n<0, 0, 2 n n!]; (* Michael Somos, Oct 20 2011 *)

PARI

{a(n) = if( n<0, 0, 2 * n * n!)}; /* Michael Somos, Oct 20 2011 */

Formula

E.g.f.: 2*x / (1 - x)^2.

Recurrence: {a(0)=0, a(1)=2, (-n^2-2*n-1)*a(n)+a(n+1)*n=0.}.

a(n) = A138770(n+2,1). - Emeric Deutsch, Apr 06 2008

a(n) = A001339(n) - A007808(n). - Michael Somos, Oct 20 2011

a(n) = (a(n-1)^2 - 2 * a(n-2)^2 + a(n-2) * a(n-3) - 4 * a(n-1) * a(n-3)) / (a(n-2) - a(n-3)) if n>2. - Michael Somos, Oct 20 2011

a(n) = 2*n*n!. - Gary Detlefs, Sep 16 2010

a(n+1) = a(n) * (n+1)^2 / n. - Reinhard Zumkeller, Nov 12 2011

0 = a(n)*(+a(n+1) -4*a(n+2) +a(n+3)) +a(n+1)*(+2*a(n+1) -a(n+3)) + a(n+2)*(+a(n+2)) if n>=0. - Michael Somos, Jun 26 2017

From Amiram Eldar, Feb 14 2021: (Start)

Sum_{n>=1} 1/a(n) = (Ei(1) - gamma)/2 = (A091725 - A001620)/2, where Ei(x) is the exponential integral.

Sum_{n>=1} (-1)^(n+1)/a(n) = (gamma - Ei(-1))/2 = (A001620 + A099285)/2. (End)

a(n) = 2 * A001563(n). - Alois P. Heinz, Sep 03 2024