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A285862 Number of permutations of [2n] with n ordered cycles such that equal-sized cycles are ordered with increasing least elements.

%I A285862 #17 May 29 2018 08:28:07
%S A285862 1,1,19,1005,62601,6061545,868380535,142349568361,27564092244689,
%T A285862 6325532235438273,1673378033771898675,505141951803309946125,
%U A285862 170002056228253072537065,63255335047795174479833625,25805276337820748477042392695,11427131417576257617280878155625
%N A285862 Number of permutations of [2n] with n ordered cycles such that equal-sized cycles are ordered with increasing least elements.
%H A285862 Alois P. Heinz, <a href="/A285862/b285862.txt">Table of n, a(n) for n = 0..200</a>
%H A285862 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A285862 a(n) = A285849(2n,n).
%e A285862 a(1) = 1: (12).
%e A285862 a(2) = 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 A285862 b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
%p A285862       (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
%p A285862       [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)))
%p A285862     end:
%p A285862 a:= n-> coeff(b(2*n$2, 0), x, n):
%p A285862 seq(a(n), n=0..20);
%t A285862 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285862 b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]]];
%t A285862 a[n_] := Coefficient[b[2n, 2n, 0], x, n];
%t A285862 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 29 2018, from Maple *)
%Y A285862 Cf. A187646, A285849, A285926.
%K A285862 nonn
%O A285862 0,3
%A A285862 _Alois P. Heinz_, Apr 27 2017