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

3, 21, 341, 9185, 343909, 16379573, 944353801, 63852563521, 4951434599465, 433032539982493, 42157340180935341, 4520992861815018433, 529496439697454958221, 67241241156753850936501, 9202393011905666532976785, 1350146007561231136610441985

Offset: 1

Author

Alois P. Heinz, Mar 02 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200
• Wikipedia, Permutation

Crossrefs

Cf. A187661, A285239, A322384.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
        b(n-i*j, min(n-i*j, i-1), max(0, t-j))/j!*
        combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
    end:
a:= n-> b(2*n$2, n)[2]:
seq(a(n), n=1..17);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0},
     Sum[Function[p, p+If[t>0 && t-j<1, {0, p[[1]]*i}, {0, 0}]][(i-1)!^j*
     b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]/j!*
     multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
a[n_] := b[2n, 2n, n][[2]];
Array[a, 17] (* Jean-François Alcover, Apr 23 2021, after Alois P. Heinz *)

Formula

a(n) = A322384(2n,n).

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