A052849 a(0) = 0; a(n) = 2*n! (n >= 1).

0, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000, 2248001455555215360000

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

For n >= 1 a(n) is the size of the centralizer of a transposition in the symmetric group S_(n+1). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 12 2001

For n > 0, a(n) = n! - A062119(n-1) = number of permutations of length n that have two specified elements adjacent. For example, a(4) = 12 as of the 24 permutations, 12 have say 1 and 2 adjacent: 1234, 2134, 1243, 2143, 3124, 3214, 4123, 4213, 3412, 3421, 4312, 4321. - Jon Perry, Jun 08 2003

With different offset, denominators of certain sums computed by Ramanujan.

From Michael Somos, Mar 04 2004: (Start)

Stirling transform of a(n) = [2, 4, 12, 48, 240, ...] is A000629(n) = [2, 6, 26, 150, 1082, ...].

Stirling transform of a(n-1) = [1, 2, 4, 12, 48, ...] is A007047(n-1) = [1, 3, 11, 51, 299, ...].

Stirling transform of a(n) = [1, 4, 12, 48, 240, ...] is A002050(n) = [1, 5, 25, 149, 1081, ...].

Stirling transform of 2*A006252(n) = [2, 2, 4, 8, 28, ...] is a(n) = [2, 4, 12, 48, 240, ...].

Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 2*A005649(n) = [4, 16, 88, 616, ...].

Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 4*A083410(n) = [4, 16, 88, 616, ...]. (End)

Number of {12, 12*, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.

Permanent of the (0, 1)-matrices with (i, j)-th entry equal to 0 if and only if it is in the border but not the corners. The border of a matrix is defined the be the first and the last row, together with the first and the last column. The corners of a matrix are the entries (i = 1, j = 1), (i = 1, j = n), (i = n, j = 1) and (i = n, j = n). - Simone Severini, Oct 17 2004

References

• B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 520.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..400
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 490.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 817.
• Anna Khmelnitskaya, Gerard van der Laan, and Dolf Talmanm, The Number of Ways to Construct a Connected Graph: A Graph-Based Generalization of the Binomial Coefficients, J. Int. Seq. (2023) Art. 23.4.3. See p. 11.
• T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
• Index entries for sequences related to factorial base representation.

Crossrefs

Essentially the same sequence as A098558.

Cf. A062119, A080046, A245334, A000142.

Row 3 of A276955 (from term a(2)=4 onward).

Programs

Haskell

a052849 n = if n == 0 then 0 else 2 * a000142 n
a052849_list = 0 : fs where fs = 2 : zipWith (*) [2..] fs
-- Reinhard Zumkeller, Aug 31 2014

Magma

[0] cat [2*Factorial(n-1): n in [2..25]]; // Vincenzo Librandi, Nov 03 2014

Maple

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

Mathematica

Join[{0}, 2Range[20]!] (* Harvey P. Dale, Jul 13 2013 *)

PARI

a(n)=if(n<1,0,n!*2)

Formula

a(n) = T(n, 2) for n>1, where T is defined as in A080046.

D-finite with recurrence: {a(0) = 0, a(1) = 2, (-1 - n)*a(n+1) + a(n+2)=0}.

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

a(n) = A090802(n, n - 1) for n > 0. - Ross La Haye, Sep 26 2005

For n >= 1, a(n) = (n+3)!*Sum_{k=0..n+2} (-1)^k*binomial(2, k)/(n + 3 - k). - Milan Janjic, Dec 14 2008

G.f.: 2/Q(0) - 2, where Q(k) = 1 - x*(k + 1)/(1 - x*(k + 1)/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Apr 01 2013

G.f.: -2 + 2/Q(0), where Q(k) = 1 + k*x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013

G.f.: W(0) - 2 , where W(k) = 1 + 1/( 1 - x*(k+1)/( x*(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013

a(n) = A245334(n, n-1), n > 0. - Reinhard Zumkeller, Aug 31 2014

From Amiram Eldar, Jan 15 2023: (Start)

Sum_{n>=1} 1/a(n) = (e-1)/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (e-1)/(2*e). (End)

Extensions

More terms from Ross La Haye, Sep 26 2005