%I M3945 #48 Mar 15 2020 14:17:59
%S 1,1,5,26,205,1936,22265,297296,4544185,78098176,1491632525,
%T 31336418816,718181418565,17831101321216,476768795646785,
%U 13658417358350336,417370516232719345,13551022195053101056
%N E.g.f.: (sin x + cos 2x) / cos 3x.
%C Arises in the enumeration of alternating 3-signed permutations.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A007286/b007286.txt">Table of n, a(n) for n = 0..200</a>
%H R. Ehrenborg and M. A. Readdy, <a href="/A006873/a006873.pdf">Sheffer posets and r-signed permutations</a>, Preprint, 1994. (Annotated scanned copy)
%H Richard Ehrenborg and Margaret A. Readdy, <a href="http://www.labmath.uqam.ca/~annales/volumes/19-2.html">Sheffer posets and r-signed permutations</a>, Annales des Sciences Mathématiques du Québec 19:2 (1995), 173-196.
%F a(n) = Re(2*((1-I)/(1+I))^n*(1+sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)*3^j))). - _Peter Luschny_, Apr 28 2013
%F a(n) ~ n! * 2^(n+1)*3^n/Pi^(n+1). - _Vaclav Kotesovec_, Jun 15 2013
%t mx = 17; Range[0, mx]! CoefficientList[ Series[ (Sin[x] + Cos[2x])/Cos[3 x], {x, 0, mx}], x] (* _Robert G. Wilson v_, Apr 28 2013 *)
%o (PARI) x='x+O('x^66); Vec(serlaplace((sin(x)+cos(2*x))/cos(3*x))) \\ _Joerg Arndt_, Apr 28 2013
%o (Sage)
%o from mpmath import mp, polylog, re
%o mp.dps = 32; mp.pretty = True
%o def aperm3(n): return 2*((1-I)/(1+I))^n*(1+add(binomial(n,j)*polylog(-j,I)*3^j for j in (0..n)))
%o def A007286(n) : return re(aperm3(n))
%o [int(A007286(n)) for n in (0..17)] # _Peter Luschny_, Apr 28 2013
%Y Cf. A006873, A007289, A225109, A002438 (bisection?).
%K nonn
%O 0,3
%A _N. J. A. Sloane_ and _Simon Plouffe_