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%I #12 Jun 01 2018 09:41:34
%S 1,23,382,5780,86029,1301673,20338679,330737236,5618265376,
%T 99849949772,1857170751804,36135886878072,734947859916792,
%U 15608257104179952,345724111468700496,7977315239656638912,191516062334747746752,4778050475554642998144,123731984754223222096512
%N Number of entries in the sixth cycles of all permutations of [n].
%C Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%H Alois P. Heinz, <a href="/A285234/b285234.txt">Table of n, a(n) for n = 6..449</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F a(n) = A185105(n,6).
%F a(n) ~ n!*n/64. - _Vaclav Kotesovec_, Apr 25 2017
%p a:= proc(n) option remember; `if`(n<7, [0$6, 1][n+1],
%p ((5*n^3-58*n^2+207*n-230)*a(n-1)-(10*n^4-152*n^3
%p +835*n^2-1973*n+1690)*a(n-2)+(n-4)*(10*n^4
%p -158*n^3+909*n^2-2251*n+2000)*a(n-3)-(5*n^6
%p -127*n^5+1330*n^4-7335*n^3+22396*n^2-35717*n
%p +23058)*a(n-4)+(n-5)^6*(n-2)*a(n-5))/((n-3)*(n-6)))
%p end:
%p seq(a(n), n=6..25);
%t b[n_, i_] := b[n, i] = Expand[If[n==0, 1, Sum[Function[p, p + Coefficient[ p, x, 0]*j*x^i][b[n-j, i+1]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]]];
%t a[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]][[6]];
%t Table[a[n], {n, 6, 25}] (* _Jean-François Alcover_, Jun 01 2018, after _Alois P. Heinz_ *)
%Y Column k=6 of A185105.
%K nonn
%O 6,2
%A _Alois P. Heinz_, Apr 15 2017
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