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A332851 Number of entries in the second cycles of all permutations of [n] when cycles are ordered by decreasing lengths.

%I A332851 #19 Dec 07 2021 22:44:01
%S A332851 1,4,21,131,950,7694,70343,709015,7867174,94945874,1241471451,
%T A332851 17442660795,262748183438,4215551220698,71873188059263,
%U A332851 1296535610096063,24690833276595022,494624740502491786,10406110949475668267,229244825016301188267,5280217980442316367510
%N A332851 Number of entries in the second cycles of all permutations of [n] when cycles are ordered by decreasing lengths.
%H A332851 Alois P. Heinz, <a href="/A332851/b332851.txt">Table of n, a(n) for n = 2..450</a>
%H A332851 Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.
%H A332851 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A332851 a(n) = Sum_{k=0..floor(n/2)} k * A349979(n,k). - _Alois P. Heinz_, Dec 07 2021
%p A332851 b:= proc(n, l) option remember; `if`(n=0, l[2], add((j-1)!*b(n-j,
%p A332851       sort([l[], j], `>`)[1..2])*binomial(n-1, j-1), j=1..n))
%p A332851     end:
%p A332851 a:= n-> b(n, [0$2]):
%p A332851 seq(a(n), n=2..23);
%t A332851 b[n_, l_] := b[n, l] = If[n == 0, l[[2]], Sum[(j-1)! b[n-j, ReverseSort[ Append[l, j]][[1 ;; 2]]] Binomial[n - 1, j - 1], {j, 1, n}]];
%t A332851 a[n_] := b[n, {0, 0}];
%t A332851 a /@ Range[2, 23] (* _Jean-François Alcover_, Mar 01 2020, after _Alois P. Heinz_ *)
%Y A332851 Column k=2 of A322384.
%Y A332851 Cf. A349979.
%K A332851 nonn
%O A332851 2,2
%A A332851 _Alois P. Heinz_, Feb 26 2020