A285233 - OEIS

%I #13 Jun 01 2018 07:59:55

%S 1,17,221,2724,34009,441383,6020276,86673088,1318681308,21194234508,

%T 359421505224,6421154849208,120637782989568,2379195625677696,

%U 49167226489281408,1062833010282628992,23992442301958329600,564697104190192569600,13836823816466433139200

%N Number of entries in the fifth 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="/A285233/b285233.txt">Table of n, a(n) for n = 5..449</a>

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

%F a(n) = A185105(n,5).

%F a(n) ~ n!*n/32. - _Vaclav Kotesovec_, Apr 25 2017

%p a:= proc(n) option remember; `if`(n<6, [0$5, 1][n+1],

%p       ((4*(n^3-9*n^2+24*n-19))*a(n-1)-(6*n^4-72*n^3+

%p        307*n^2-547*n+334)*a(n-2)+(4*n^5-64*n^4+398*n^3

%p       -1191*n^2+1683*n-862)*a(n-3)-(n-4)^5*(n-1)*a(n-4))

%p       /((n-2)*(n-5)))

%p     end:

%p seq(a(n), n=5..25);

%t a[3] = a[4] = 0; a[5] = 1; a[6] = 17; a[n_] := a[n] = ((4(n^3 - 9n^2 + 24n - 19)) a[n-1] - (6n^4 - 72n^3 + 307n^2 - 547n + 334) a[n-2] + (4n^5 - 64n^4 + 398n^3 - 1191n^2 + 1683n - 862) a[n-3] - (n-4)^5 (n-1) a[n-4]) / ((n - 2)(n - 5));

%t Table[a[n], {n, 5, 25}] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)

%Y Column k=5 of A185105.

%K nonn

%O 5,2

%A _Alois P. Heinz_, Apr 15 2017