A000704 Number of degree-n even permutations of order dividing 2.

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

Author

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

Keywords

• nonn
• easy

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.

Programs

Magma

m:=30; R:=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

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 *)

PARI

my(x='x+O('x^30)); Vec(serlaplace( exp(x)*cosh(x^2/2) )) \\ 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

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

Extensions

More terms from Harvey P. Dale, Nov 29 2013