A285235 - OEIS

%I #13 Jun 01 2018 08:04:29

%S 1,30,622,11378,199809,3499572,62333543,1141073295,21593291506,

%T 423749322362,8637159909596,182967605341204,4028364756058464,

%U 92147187469290768,2188667860854515856,53939340317601471888,1378181549321980128288,36476226109960185948768

%N Number of entries in the seventh 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="/A285235/b285235.txt">Table of n, a(n) for n = 7..450</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

%F a(n) = A185105(n,7).

%F 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 a(n) ~ n!*n/128. - _Vaclav Kotesovec_, Apr 25 2017

%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,

%p       add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(

%p       b(n-j, max(0, i-1)))*binomial(n-1, j-1)*

%p       (j-1)!, j=1..n)))

%p     end:

%p a:= n-> coeff(b(n, 7), x, 1):

%p seq(a(n), n=7..30);

%t 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 a[n_] := Coefficient[b[n, 7], x, 1];

%t Table[a[n], {n, 7, 30}] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)

%Y Column k=7 of A185105.

%K nonn

%O 7,2

%A _Alois P. Heinz_, Apr 15 2017