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A285853 Number of permutations of [n] with two ordered cycles such that equal-sized cycles are ordered with increasing least elements.

%I A285853 #17 Apr 29 2017 18:53:48
%S A285853 1,6,19,100,508,3528,24876,219168,1980576,21257280,234434880,
%T A285853 2972885760,38715943680,566931294720,8514866707200,141468564787200,
%U A285853 2407290355814400,44753976117043200,850965783594393600,17505896073523200000,367844990453821440000
%N A285853 Number of permutations of [n] with two ordered cycles such that equal-sized cycles are ordered with increasing least elements.
%H A285853 Alois P. Heinz, <a href="/A285853/b285853.txt">Table of n, a(n) for n = 2..450</a>
%H A285853 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A285853 a(2) = 1: (1)(2).
%e A285853 a(3) = 6: (1)(23), (23)(1), (2)(13), (13)(2), (3)(12), (12)(3).
%e A285853 a(4) = 19: (123)(4), (4)(123), (132)(4), (4)(132), (124)(3), (3)(124), (142)(3), (3)(142), (134)(2), (2)(134), (143)(2), (2)(143), (1)(234), (234)(1), (1)(243), (243)(1),  (12)(34), (13)(24), (14)(23).
%p A285853 a:= n-> 2*add(binomial(n, k)*(k-1)!*(n-k-1)!, k=1..n/2)-
%p A285853         `if`(n::even, 3/2*binomial(n, n/2)*(n/2-1)!^2, 0):
%p A285853 seq(a(n), n=2..25);
%p A285853 # second Maple program:
%p A285853 a:= proc(n) option remember; `if`(n<5, [0, 1, 6, 19][n],
%p A285853      ((2*n-1)*(n-1)*a(n-1)+(n-2)*(2*n^2-5*n-1)*a(n-2)
%p A285853       -(n-3)^2*((2*n^2-5*n+4)*a(n-3)+(n-4)^2*a(n-4)))/(2*n))
%p A285853     end:
%p A285853 seq(a(n), n=2..25);
%t A285853 Table[(n-1)!*(2*HarmonicNumber[n] - (3 + (-1)^n)/n), {n, 2, 25}] (* _Vaclav Kotesovec_, Apr 29 2017 *)
%Y A285853 Column k=2 of A285849.
%Y A285853 Cf. A285917.
%K A285853 nonn
%O A285853 2,2
%A A285853 _Alois P. Heinz_, Apr 27 2017