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A300931 a(n) is the number of Klein four orbits of permutations generated by the operations of inverse permutation and conjugation by w_0 = [n, n-1, ..., 1].

%I A300931 #27 May 19 2020 06:28:42
%S A300931 1,1,2,4,13,45,230,1388,10558,92126,912908,9998008,119831996,
%T A300931 1557050972,21795929320,326923928048,5230723155848,88921965504136,
%U A300931 1600593971537552,30411277553507360,608225514852464848,12772735603832679248,281000182274281641056
%N A300931 a(n) is the number of Klein four orbits of permutations generated by the operations of inverse permutation and conjugation by w_0 = [n, n-1, ..., 1].
%H A300931 Alois P. Heinz, <a href="/A300931/b300931.txt">Table of n, a(n) for n = 0..450</a>
%F A300931 a(n) = (n! + (2*floor(n/2))!! + 2 + 2*Sum_{k=1..floor(n/2)} (Product_{j=0..k-1} binomial(n-2j,2))/k!)/4.
%F A300931 For n > 1, a(n) = (n! + (2*floor(n/2))!!)/4 + A001475(n-1). - _Vaclav Kotesovec_, May 19 2020
%e A300931 For n=3, the a(3)=4 orbits are {(1,2,3)}, {(1,3,2),(2,1,3)}, {(2,3,1),(3,1,2)}, and {(3,2,1)}.
%p A300931 a:= proc(n) option remember; `if`(n<6, [1$2, 2, 4, 13, 45][n+1],
%p A300931       ((15*n^2+11*n-161)*a(n-1)-(19*n^2-100*n+65)*a(n-2)
%p A300931        -(30*n^3-87*n^2-81*n+266)*a(n-3)+68*(n-2)*(n-3)^2*a(n-4)
%p A300931        +(n-3)*(n-4)*(15*n^2-23*n-16)*a(n-5)-2*(17*n-33)*(n-3)*
%p A300931        (n-4)*(n-5)*a(n-6))/(15*n-38))
%p A300931     end:
%p A300931 seq(a(n), n=0..35);  # _Alois P. Heinz_, Mar 30 2018
%t A300931 Table[(n! + (2 Floor[n/2])!! + 2 + 2 Sum[Product[Binomial[n - 2 j, 2], {j, 0, k - 1}]/k!, {k, Floor[n/2]}])/4, {n, 22}] (* _Michael De Vlieger_, Mar 16 2018 *)
%t A300931 Table[(n! + (2*Floor[n/2])!!)/4 + I^(1 - n) * 2^((n - 3)/2) * HypergeometricU[(1 - n)/2, 3/2, -1/2], {n, 0, 25}] (* _Vaclav Kotesovec_, May 19 2020 *)
%Y A300931 Cf. A001475.
%K A300931 nonn
%O A300931 0,3
%A A300931 _Sara Billey_, Philippe Nadeau, _Jordan Weaver_, Jesse Rivera, Justin Shyi, Mar 15 2018
%E A300931 a(19)-a(22) from _Michael De Vlieger_, Mar 16 2018
%E A300931 a(0)=1 prepended by _Alois P. Heinz_, Mar 30 2018