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%I #8 Apr 21 2020 08:30:35
%S 1,21,226,2612,29261,346453,4338214,57819554,815225643,12234293579,
%T 194294281572,3264124624256,57826690252441,1079032037759257,
%U 21142347350725466,434563256137908638,9344589765620199919,209952915324112384719,4919186923210370523448
%N Number of entries in the fourth cycles of all permutations of [n] when cycles are ordered by increasing lengths.
%H Alois P. Heinz, <a href="/A332908/b332908.txt">Table of n, a(n) for n = 4..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>
%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, 4)[2]:
%p seq(a(n), n=4..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, 4][[2]];
%t a /@ Range[4, 22] (* _Jean-François Alcover_, Apr 21 2020, after _Alois P. Heinz_ *)
%Y Column k=4 of A322383.
%K nonn
%O 4,2
%A _Alois P. Heinz_, Mar 02 2020
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