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

%I A332906 #11 Dec 07 2021 21:49:08
%S A332906 1,7,37,241,1661,13301,117209,1150297,12314329,144593989,1828734689,
%T A332906 24995387561,365311053953,5707795873261,94637770625761,
%U A332906 1665132643843201,30896642665904609,604541044692565157,12416248460455779089,267500866283111679289,6024053249628809274769
%N A332906 Number of entries in the second cycles of all permutations of [n] when cycles are ordered by increasing lengths.
%H A332906 Alois P. Heinz, <a href="/A332906/b332906.txt">Table of n, a(n) for n = 2..450</a>
%H A332906 Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.
%H A332906 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A332906 a(n) = Sum_{k>=0} k * A349980(n,k). - _Alois P. Heinz_, Dec 07 2021
%p A332906 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
%p A332906       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
%p A332906         b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
%p A332906          (n, i$j, n-i*j)), j=0..n/i)))
%p A332906     end:
%p A332906 a:= n-> b(n, 1, 2)[2]:
%p A332906 seq(a(n), n=2..22);
%t A332906 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A332906 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}]]];
%t A332906 a[n_] := b[n, 1, 2][[2]];
%t A332906 a /@ Range[2, 22] (* _Jean-François Alcover_, Apr 21 2020, after _Alois P. Heinz_ *)
%Y A332906 Column k=2 of A322383.
%Y A332906 Cf. A349980.
%K A332906 nonn
%O A332906 2,2
%A A332906 _Alois P. Heinz_, Mar 02 2020