A000704 Number of degree-n even permutations of order dividing 2. (Formerly M3511 N1427)

1, 1, 1, 1, 4, 16, 46, 106, 316, 1324, 5356, 18316, 63856, 272416, 1264264, 5409496, 22302736, 101343376, 507711376, 2495918224, 11798364736, 58074029056, 309240315616, 1670570920096, 8792390355904, 46886941456576, 264381946998976, 1533013006902976

Offset

0,5

Comments

Number of odd partitions of an n-element set avoiding the pattern 123 (see Goyt paper). - Ralf Stephan, May 08 2007

References

J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, Inc. New York, 1958 (Chap. 4, Problem 22).

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

T. D. Noe, Table of n, a(n) for n=0..200

Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.

A. M. Goyt, Avoidance of partitions of a 3-element set, arXiv:math/0603481 [math.CO], 2006-2007.

L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Formula

E.g.f.: exp(x)*cosh(x^2/2).

a(n) = Sum_{i = 0..floor(n/4)} C(n, 4i)*(4i-1)!!. - Ralf Stephan, May 08 2007 [Corrected by Sean A. Irvine, Mar 01 2011]

Conjecture: a(n) -3*a(n-1) +3*a(n-2) -a(n-3) -(n-1)*(n-3)*a(n-4) +(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 03 2014

Mathematica

a[n_] := Sum[(4i - 1)!! Binomial[n, 4i], {i, 0, n/4}]; Array[a, 30, 0] (* Robert G. Wilson v *)
With[{nn = 30}, CoefficientList[Series[Exp[x]Cosh[x^2/2], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Nov 29 2013 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x)*cosh(x^2/2) )) \\ G. C. Greubel, Jul 02 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x)*Cosh(x^2/2) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 02 2019
(Sage) m = 30; T = taylor(exp(x)*cosh(x^2/2), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jul 02 2019

Crossrefs

Sequence in context: A213480 A306302 A159940 * A374320 A374262 A007315

Adjacent sequences: A000701 A000702 A000703 * A000705 A000706 A000707

Keyword

nonn,easy

Author

N. J. A. Sloane and J. H. Conway

Extensions

More terms from Harvey P. Dale, Nov 29 2013

Status

approved