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A285231 Number of entries in the third cycles of all permutations of [n].

%I A285231 #16 May 30 2018 06:07:25
%S A285231 1,8,59,463,3978,37566,388728,4385592,53653680,708126480,10034314560,
%T A285231 152001161280,2451821339520,41964428419200,759698874547200,
%U A285231 14505012898790400,291323663566387200,6140173922952652800,135515391451776000000,3125606951427609600000
%N A285231 Number of entries in the third cycles of all permutations of [n].
%C A285231 Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%H A285231 Alois P. Heinz, <a href="/A285231/b285231.txt">Table of n, a(n) for n = 3..449</a>
%H A285231 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A285231 a(n) = A185105(n,3) = A159324(n-1)/2.
%F A285231 a(n) ~ n!*n/8. - _Vaclav Kotesovec_, Apr 25 2017
%p A285231 a:= proc(n) option remember; `if`(n<4, [0$3, 1][n+1],
%p A285231       ((2*n^3-7*n^2+3*n+4)*a(n-1)-(n-2)^3*(n+1)*a(n-2))/(n*(n-3)))
%p A285231     end:
%p A285231 seq(a(n), n=0..25);
%t A285231 a[3] = 1; a[4] = 8;
%t A285231 a[n_] := a[n] = ((2n^3-7n^2+3n+4) a[n-1] - (n-2)^3 (n+1) a[n-2])/(n(n-3));
%t A285231 Table[a[n], {n, 3, 25}] (* _Jean-François Alcover_, May 30 2018, from Maple *)
%Y A285231 Column k=3 of A185105.
%Y A285231 Cf. A159324.
%K A285231 nonn
%O A285231 3,2
%A A285231 _Alois P. Heinz_, Apr 15 2017