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A285235 Number of entries in the seventh cycles of all permutations of [n].

%I A285235 #13 Jun 01 2018 08:04:29
%S A285235 1,30,622,11378,199809,3499572,62333543,1141073295,21593291506,
%T A285235 423749322362,8637159909596,182967605341204,4028364756058464,
%U A285235 92147187469290768,2188667860854515856,53939340317601471888,1378181549321980128288,36476226109960185948768
%N A285235 Number of entries in the seventh cycles of all permutations of [n].
%C A285235 Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%H A285235 Alois P. Heinz, <a href="/A285235/b285235.txt">Table of n, a(n) for n = 7..450</a>
%H A285235 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A285235 a(n) = A185105(n,7).
%F A285235 Recurrence: (n-7)*(n-4)*a(n) = (n-3)*(6*n^2 - 67*n + 176)*a(n-1) - 5*(n-4)*(3*n^3 - 43*n^2 + 195*n - 283)*a(n-2) + 10*(2*n^5 - 47*n^4 + 436*n^3 - 1999*n^2 + 4532*n - 4062)*a(n-3) - (15*n^6 - 445*n^5 + 5465*n^4 - 35555*n^3 + 129161*n^2 - 248111*n + 196528)*a(n-4) + (6*n^7 - 221*n^6 + 3473*n^5 - 30165*n^4 + 156251*n^3 - 482105*n^2 + 819087*n - 589808)*a(n-5) - (n-6)^7*(n-3)*a(n-6), for n>7. - _Vaclav Kotesovec_, Apr 25 2017
%F A285235 a(n) ~ n!*n/128. - _Vaclav Kotesovec_, Apr 25 2017
%p A285235 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A285235       add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(
%p A285235       b(n-j, max(0, i-1)))*binomial(n-1, j-1)*
%p A285235       (j-1)!, j=1..n)))
%p A285235     end:
%p A285235 a:= n-> coeff(b(n, 7), x, 1):
%p A285235 seq(a(n), n=7..30);
%t A285235 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 A285235 a[n_] := Coefficient[b[n, 7], x, 1];
%t A285235 Table[a[n], {n, 7, 30}] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)
%Y A285235 Column k=7 of A185105.
%K A285235 nonn
%O A285235 7,2
%A A285235 _Alois P. Heinz_, Apr 15 2017