A002484 Number of ménage permutations. (Formerly M1524 N0597)

1, 2, 5, 20, 87, 616, 4843, 44128, 444621, 4936274, 59661265, 780547332, 10987097799, 165587196328, 2660378564791, 45392026278108, 819716784789209, 15620011000052754, 313219935456572497, 6593238656843759572

Offset

3,2

References

C. Berge, Principles of Combinatorics, Academic Press, NY, 1971, p. 162.

E. N. Gilbert, Knots and classes of menage permutations, Scripta Math., 22 (1956), 228-233 (1957).

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 195.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 3..200

E. N. Gilbert, Knots and classes of menage permutations [Annotated scanned copy of preprint]

Formula

Gilbert gives a formula (see Maple code).

a(n) ~ (n-1)! * exp(-2). - Vaclav Kotesovec, May 23 2014

Maple

with(numtheory): d := n->divisors(n): U := (m, t)->sum(2*m*binomial(2*m-k, k)*(m-k)!*(t-1)^k/(2*m-k), k=0..m): A := (n, i)->phi(n/dd[i])*(n/dd[i])^dd[i]*U(dd[i], 1-dd[i]/n)/n: for n from 3 to 28 do dd := d(n): B := [seq(A(n, j), j=1..nops(dd))]: a[n] := sum(B[i], i=1..nops(B)) od: seq(a[n], n=3..28); # Emeric Deutsch, Mar 08 2004

Mathematica

u[m_, t_] := Sum[ 2m*Binomial[ 2m-k, k]*(m-k)!*((t-1)^k / (2m-k)), {k, 0, m}]; a[n_] := Sum[ EulerPhi[n/d] * (n/d)^d * (u[d, 1-d/n]/n), {d, Divisors[n]} ]; Table[ a[n], {n, 3, 22} ] (* Jean-François Alcover, Dec 07 2011, after Maple *)

Crossrefs

Cf. A000179.

Sequence in context: A262171 A262172 A258830 * A280102 A366452 A277686

Adjacent sequences: A002481 A002482 A002483 * A002485 A002486 A002487

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Mar 08 2004

Status

approved