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A332854 Number of entries in the fifth cycles of all permutations of [n] when cycles are ordered by decreasing lengths.

%I A332854 #8 Mar 13 2021 10:04:34
%S A332854 1,16,197,2311,27568,343909,4541329,63719723,949770615,15010838233,
%T A332854 250997692441,4430433962701,82376202579421,1610014961936672,
%U A332854 33010385435710028,708642421376230354,15899009565671538930,372166745683645768206,9074749796104015627262
%N A332854 Number of entries in the fifth cycles of all permutations of [n] when cycles are ordered by decreasing lengths.
%H A332854 Alois P. Heinz, <a href="/A332854/b332854.txt">Table of n, a(n) for n = 5..450</a>
%H A332854 Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.
%H A332854 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%p A332854 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A332854       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
%p A332854         b(n-i*j, min(n-i*j, i-1), max(0, t-j))/j!*
%p A332854         combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
%p A332854     end:
%p A332854 a:= n-> b(n$2, 5)[2]:
%p A332854 seq(a(n), n=5..23);
%t A332854 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A332854 b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i<1, {0, 0},
%t A332854      Sum[Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i},
%t A332854      {0, 0}]][(i-1)!^j*b[n - i*j, Min[n - i*j, i-1], Max[0, t-j]]/j!*
%t A332854      multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
%t A332854 a[n_] := b[n, n, 5][[2]];
%t A332854 Table[a[n], {n, 5, 23}] (* _Jean-François Alcover_, Mar 13 2021, after _Alois P. Heinz_ *)
%Y A332854 Column k=5 of A322384.
%K A332854 nonn
%O A332854 5,2
%A A332854 _Alois P. Heinz_, Feb 26 2020