%I #26 Jan 03 2021 15:16:55
%S 1,3,9,45,270,1680,11340,108108,1005480,10929600,114845445,1543647105,
%T 20367547200,305087041350,4428663384600,76921682828760,
%U 1322260935065280,24987504206574000,464475474578648925,9916256376966427425,209858853988540310400,4809961676643673161150
%N Number of even permutations on n letters that have a root.
%H Lev Glebsky, Melany Licón, and 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 M. R. Pournaki, <a href="https://ajc.maths.uq.edu.au/pdf/45/ajc_v45_p037.pdf">On the number of even permutations with roots</a>, Australasian Journal of Combinatorics, Volume 45 (2009), Pages 37-42. See Theorem 1.1 p. 37.
%F E.g.f.: sqrt((1+x)/(1-x))*Product_{k>=1} cosh(x^(2*k)/(2*k)) - (1/2)*Product_{k>=1} (1+x^(2*k-1)/(2*k-1)) * (Product_{k>=1} cosh(x^(2*k)/(2*k)) - Product_{k>=1} cos(x^(2*k)/(2*k))) for n >= 2.
%t m = 24;
%t (Sqrt[(1+x)/(1-x)] Product[Cosh[x^(2k)/(2k)], {k, 1, m}] - (1/2) Product[1 + x^(2k-1)/(2k-1), {k, 1, m}] (Product[Cosh[x^(2k)/(2k)], {k, 1, m}] - Product[Cos[x^(2k)/(2k)], {k, 1, m}]) + O[x]^m // CoefficientList[#, x]&)Range[0, m-1]! // Drop[#, 2]& (* _Jean-François Alcover_, Nov 17 2020 *)
%o (PARI) my(N=30, x='x+O('x^N), v=Vec(serlaplace(sqrt((1+x)/(1-x))*prod(k=1, N, cosh(x^(2*k)/(2*k))) - (1/2)*prod(k=1, N, 1+x^(2*k-1)/(2*k-1))*(prod(k=1, N, cosh(x^(2*k)/(2*k))) - prod(k=1, N, cos(x^(2*k)/(2*k))))))); vector(#v-2, k, v[k+2])
%Y Cf. A003483.
%K nonn
%O 2,2
%A _Michel Marcus_, Nov 04 2020