A338575 Number of even permutations on n letters that have a root.

1, 3, 9, 45, 270, 1680, 11340, 108108, 1005480, 10929600, 114845445, 1543647105, 20367547200, 305087041350, 4428663384600, 76921682828760, 1322260935065280, 24987504206574000, 464475474578648925, 9916256376966427425, 209858853988540310400, 4809961676643673161150

Offset

2,2

Links

Table of n, a(n) for n=2..23.

Lev Glebsky, Melany Licón, and Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.

M. R. Pournaki, On the number of even permutations with roots, Australasian Journal of Combinatorics, Volume 45 (2009), Pages 37-42. See Theorem 1.1 p. 37.

Formula

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.

Mathematica

m = 24;
(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 *)

Prog

(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])

Crossrefs

Cf. A003483.

Sequence in context: A038059 A369388 A174318 * A004990 A027616 A245140

Adjacent sequences: A338572 A338573 A338574 * A338576 A338577 A338578

Keyword

nonn

Author

Michel Marcus, Nov 04 2020

Status

approved