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

1,1

Links

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

Wikipedia, Permutation

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

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 *)

Crossrefs

Cf. A187661, A285239, A322384.

Sequence in context: A113085 A240936 A342245 * A083228 A052445 A351130

Adjacent sequences: A332925 A332926 A332927 * A332929 A332930 A332931

Keyword

nonn

Author

Alois P. Heinz, Mar 02 2020

Status

approved