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%I M3511 N1427 #47 Sep 08 2022 08:44:28
%S 1,1,1,1,4,16,46,106,316,1324,5356,18316,63856,272416,1264264,5409496,
%T 22302736,101343376,507711376,2495918224,11798364736,58074029056,
%U 309240315616,1670570920096,8792390355904,46886941456576,264381946998976,1533013006902976
%N Number of degree-n even permutations of order dividing 2.
%C Number of odd partitions of an n-element set avoiding the pattern 123 (see Goyt paper). - _Ralf Stephan_, May 08 2007
%D J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, Inc. New York, 1958 (Chap. 4, Problem 22).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000704/b000704.txt">Table of n, a(n) for n=0..200</a>
%H Lev Glebsky, Melany Licón, Luis Manuel Rivera, <a href="https://arxiv.org/abs/1907.00548">On the number of even roots of permutations</a>, arXiv:1907.00548 [math.CO], 2019.
%H A. M. Goyt, <a href="https://arxiv.org/abs/math/0603481">Avoidance of partitions of a 3-element set</a>, arXiv:math/0603481 [math.CO], 2006-2007.
%H L. Moser and M. Wyman, <a href="https://doi.org/10.4153/CJM-1955-020-0">On solutions of x^d = 1 in symmetric groups</a>, Canad. J. Math., 7 (1955), 159-168.
%F E.g.f.: exp(x)*cosh(x^2/2).
%F 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]
%F 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
%t a[n_] := Sum[(4i - 1)!! Binomial[n, 4i], {i, 0, n/4}]; Array[a, 30, 0] (* _Robert G. Wilson v_ *)
%t With[{nn = 30}, CoefficientList[Series[Exp[x]Cosh[x^2/2], {x, 0, nn}], x] Range[0, nn]!] (* _Harvey P. Dale_, Nov 29 2013 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x)*cosh(x^2/2) )) \\ _G. C. Greubel_, Jul 02 2019
%o (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
%o (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
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_ and _J. H. Conway_
%E More terms from _Harvey P. Dale_, Nov 29 2013
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