A332906 - OEIS

%I #11 Dec 07 2021 21:49:08

%S 1,7,37,241,1661,13301,117209,1150297,12314329,144593989,1828734689,

%T 24995387561,365311053953,5707795873261,94637770625761,

%U 1665132643843201,30896642665904609,604541044692565157,12416248460455779089,267500866283111679289,6024053249628809274769

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

%H Alois P. Heinz, <a href="/A332906/b332906.txt">Table of n, a(n) for n = 2..450</a>

%H Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

%F a(n) = Sum_{k>=0} k * A349980(n,k). - _Alois P. Heinz_, Dec 07 2021

%p b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,

%p       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*

%p         b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]

%p          (n, i$j, n-i*j)), j=0..n/i)))

%p     end:

%p a:= n-> b(n, 1, 2)[2]:

%p seq(a(n), n=2..22);

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t 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 a[n_] := b[n, 1, 2][[2]];

%t a /@ Range[2, 22] (* _Jean-François Alcover_, Apr 21 2020, after _Alois P. Heinz_ *)

%Y Column k=2 of A322383.

%Y Cf. A349980.

%K nonn

%O 2,2

%A _Alois P. Heinz_, Mar 02 2020