A332906 Number of entries in the second cycles of all permutations of [n] when cycles are ordered by increasing lengths.

1, 7, 37, 241, 1661, 13301, 117209, 1150297, 12314329, 144593989, 1828734689, 24995387561, 365311053953, 5707795873261, 94637770625761, 1665132643843201, 30896642665904609, 604541044692565157, 12416248460455779089, 267500866283111679289, 6024053249628809274769

Offset: 2

Author

Alois P. Heinz, Mar 02 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..450
• Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
• Wikipedia, Permutation

Crossrefs

Column k=2 of A322383.

Cf. A349980.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
        b(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(n, 1, 2)[2]:
seq(a(n), n=2..22);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i>n, 0, Sum[Function[ p, p + If[p =!= 0 && t>0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][(i-1)!^j* b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Array[i&, j], n - i*j]]], {j, 0, n/i}]]];
a[n_] := b[n, 1, 2][[2]];
a /@ Range[2, 22] (* Jean-François Alcover, Apr 21 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k>=0} k * A349980(n,k). - Alois P. Heinz, Dec 07 2021