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A285239 Number of entries in the n-th cycles of all permutations of [2n].

%I A285239 #17 Mar 10 2020 04:03:17
%S A285239 3,27,463,12217,441383,20338679,1141073295,75473055841,5748862140283,
%T A285239 495446888127507,47648289796265871,5057570671179281161,
%U A285239 587173799850231036207,74005641366738437835967,10062023872139208015273375,1467822867614662009540883265
%N A285239 Number of entries in the n-th cycles of all permutations of [2n].
%C A285239 Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%C A285239 All terms are odd.
%H A285239 Alois P. Heinz, <a href="/A285239/b285239.txt">Table of n, a(n) for n = 1..326</a>
%H A285239 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A285239 a(n) = A185105(2n,n).
%F A285239 a(n) ~ 2^(3*n-1) * c^(2*n) * n^(n - 1/2) / (sqrt(Pi*(c-1)) * (2*c-1)^n * exp(n)), where c = -LambertW(-1,-exp(-1/2)/2) = 1.7564312086261696769827376166... - _Vaclav Kotesovec_, Apr 15 2017, updated Mar 10 2020
%p A285239 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A285239       add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))((j-1)!
%p A285239       *b(n-j, max(0, i-1)))*binomial(n-1, j-1), j=1..n)))
%p A285239     end:
%p A285239 a:= n-> coeff(b(2*n, n), x, 1):
%p A285239 seq(a(n), n=1..20);
%t A285239 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, Sum[Function[p, p + If[i == 1, Coefficient[p, x, 0] j x, 0]][(j - 1)! b[n - j, Max[0, i - 1]]] Binomial[ n - 1, j - 1], {j, 1, n}]]];
%t A285239 a[n_] := Coefficient[b[2n, n], x, 1];
%t A285239 Array[a, 20] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)
%Y A285239 Cf. A185105.
%K A285239 nonn
%O A285239 1,1
%A A285239 _Alois P. Heinz_, Apr 15 2017