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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Harvey P. Dale, Nov 29 2013
STATUS
approved