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A332928 Number of entries in the n-th cycles of all permutations of [2n] when cycles are ordered by decreasing lengths.

%I A332928 #15 Apr 23 2021 08:20:33
%S A332928 3,21,341,9185,343909,16379573,944353801,63852563521,4951434599465,
%T A332928 433032539982493,42157340180935341,4520992861815018433,
%U A332928 529496439697454958221,67241241156753850936501,9202393011905666532976785,1350146007561231136610441985
%N A332928 Number of entries in the n-th cycles of all permutations of [2n] when cycles are ordered by decreasing lengths.
%H A332928 Alois P. Heinz, <a href="/A332928/b332928.txt">Table of n, a(n) for n = 1..200</a>
%H A332928 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A332928 a(n) = A322384(2n,n).
%F A332928 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_, Mar 10 2020
%p A332928 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A332928       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
%p A332928         b(n-i*j, min(n-i*j, i-1), max(0, t-j))/j!*
%p A332928         combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
%p A332928     end:
%p A332928 a:= n-> b(2*n$2, n)[2]:
%p A332928 seq(a(n), n=1..17);
%t A332928 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A332928 b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0},
%t A332928      Sum[Function[p, p+If[t>0 && t-j<1, {0, p[[1]]*i}, {0, 0}]][(i-1)!^j*
%t A332928      b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]/j!*
%t A332928      multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
%t A332928 a[n_] := b[2n, 2n, n][[2]];
%t A332928 Array[a, 17] (* _Jean-François Alcover_, Apr 23 2021, after _Alois P. Heinz_ *)
%Y A332928 Cf. A187661, A285239, A322384.
%K A332928 nonn
%O A332928 1,1
%A A332928 _Alois P. Heinz_, Mar 02 2020