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A285236 Number of entries in the eighth cycles of all permutations of [n].

%I A285236 #13 Jun 01 2018 08:05:48
%S A285236 1,38,964,21018,431007,8671656,175065071,3591984289,75473055841,
%T A285236 1631318215818,36369569578502,837619857754240,19943142053389024,
%U A285236 491010028537071248,12499878460133012064,328936666440527737296,8943724877454118086096,251125623168859020015072
%N A285236 Number of entries in the eighth cycles of all permutations of [n].
%C A285236 Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%H A285236 Alois P. Heinz, <a href="/A285236/b285236.txt">Table of n, a(n) for n = 8..450</a>
%H A285236 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A285236 a(n) = A185105(n,8).
%F A285236 Recurrence: (n-8)*(n-5)*a(n) = (7*n^3 - 117*n^2 + 618*n - 1036)*a(n-1) - (21*n^4 - 450*n^3 + 3521*n^2 - 11996*n + 15092)*a(n-2) + 5*(7*n^5 - 190*n^4 + 2039*n^3 - 10842*n^2 + 28614*n - 30016)*a(n-3) - (35*n^6 - 1185*n^5 + 16635*n^4 - 124015*n^3 + 518011*n^2 - 1149493*n + 1058400)*a(n-4) + (n-6)*(21*n^6 - 747*n^5 + 11033*n^4 - 86597*n^3 + 380805*n^2 - 888917*n + 859586)*a(n-5) - (7*n^8 - 352*n^7 + 7728*n^6 - 96726*n^5 + 754656*n^4 - 3756732*n^3 + 11646888*n^2 - 20547489*n + 15780868)*a(n-6) + (n-7)^8*(n-4)*a(n-7), for n>8. - _Vaclav Kotesovec_, Apr 25 2017
%F A285236 a(n) ~ n!*n/256. - _Vaclav Kotesovec_, Apr 25 2017
%p A285236 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A285236       add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(
%p A285236       b(n-j, max(0, i-1)))*binomial(n-1, j-1)*
%p A285236       (j-1)!, j=1..n)))
%p A285236     end:
%p A285236 a:= n-> coeff(b(n, 8), x, 1):
%p A285236 seq(a(n), n=8..30);
%t A285236 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, Sum[Function[p, p + If[i == 1, Coefficient[p, x, 0]*j*x, 0]][b[n - j, Max[0, i - 1]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]]];
%t A285236 a[n_] := Coefficient[b[n, 8], x, 1];
%t A285236 Table[a[n], {n, 8, 30}] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)
%Y A285236 Column k=8 of A185105.
%K A285236 nonn
%O A285236 8,2
%A A285236 _Alois P. Heinz_, Apr 15 2017