A032184 a(n) = 2^n*(n-1)! for n > 1, a(1) = 1.

1, 4, 16, 96, 768, 7680, 92160, 1290240, 20643840, 371589120, 7431782400, 163499212800, 3923981107200, 102023508787200, 2856658246041600, 85699747381248000, 2742391916199936000, 93241325150797824000, 3356687705428721664000, 127554132806291423232000

Offset

1,2

Comments

Previous name was: "CIJ" (necklace, indistinct, labeled) transform of 1, 3, 5, 7, ...

Links

Table of n, a(n) for n=1..20.

C. G. Bower, Transforms (2).

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 565.

Index entries for sequences related to necklaces.

Formula

a(n) = 2^n*(n-1)! for n > 1, a(1) = 1.

E.g.f.: (1 + 2*x)/(1 - 2*x). - Paul Barry, May 26 2003 [This e.g.f. yields the sequence (a(n+1): n >= 0). - M. F. Hasler, Jan 15 2017]

a(n) + 2*(-n+1)*a(n-1) = 0. - R. J. Mathar, Nov 30 2012 [Valid for n >= 3; equivalently: a(n+1) = 2*n*a(n) for n > 1. - M. F. Hasler, Jan 15 2017]

G.f.: G(0) - 1, where G(k) = 1 + 1/(1 - 1/(1 + 1/((2*k + 2)*x*G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013

Let s(n) = Sum_{k >= 1} 1/(2*k - 1)^n with n > 1, then s(n) = (-1)^n*PolyGamma(n-1, 1/2)/a(n). - Jean-François Alcover, Dec 18 2013

a(n) = round(-zeta(n)(1/2)) where zeta(n)(1/2) is the n-th derivative of the zeta function at 1/2. - Artur Jasinski, Feb 06 2021

E.g.f.: -x-log(1-2*x). - Alois P. Heinz, Mar 10 2022

Sum_{n>=1} 1/a(n) = (exp(1/2)+1)/2. - Amiram Eldar, Feb 02 2023

Maple

A032184:=n->if n>1 then 2^n*(n-1)! else 1 fi: seq(A032184(n), n=1..30); # Wesley Ivan Hurt and M. F. Hasler, Jan 15 2017

Mathematica

Join[{1}, Table[2^n (n-1)!, {n, 2, 20}]] (* Harvey P. Dale, Oct 08 2017 *)

Prog

(PARI) apply( A032184=n->(n-1)!<<n-(n==1) , [1..18]) \\ M. F. Hasler, Jan 15 2017

Crossrefs

Apart from the initial term, same as A066318.

Sequence in context: A317998 A027745 A293143 * A130683 A111976 A236772

Adjacent sequences: A032181 A032182 A032183 * A032185 A032186 A032187

Keyword

nonn,easy

Author

Christian G. Bower

Extensions

New name (using first formula) from Joerg Arndt, Mar 10 2022

Status

approved