cp's OEIS Frontend

A285238 Number of entries in the tenth cycles of all permutations of [n].

%I A285238 #13 Jun 01 2018 08:10:29
%S A285238 1,57,2061,61524,1672323,43426821,1106667572,28127644296,720378419177,
%T A285238 18715673685469,495446888127507,13403690294272704,371315688867567088,
%U A285238 10546557230068193568,307378160401299252032,9196581430595518185328,282526394585486139996736
%N A285238 Number of entries in the tenth cycles of all permutations of [n].
%C A285238 Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%H A285238 Alois P. Heinz, <a href="/A285238/b285238.txt">Table of n, a(n) for n = 10..450</a>
%H A285238 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A285238 a(n) = A185105(n,10).
%F A285238 Recurrence: (n-10)*(n-7)*a(n) = (9*n^3 - 196*n^2 + 1361*n - 3006)*a(n-1) - 2*(18*n^4 - 496*n^3 + 5001*n^2 - 21971*n + 35682)*a(n-2) + 14*(6*n^5 - 206*n^4 + 2797*n^3 - 18829*n^2 + 63002*n - 83988)*a(n-3) - 7*(18*n^6 - 758*n^5 + 13240*n^4 - 122950*n^3 + 640883*n^2 - 1779393*n + 2057142)*a(n-4) + 7*(18*n^7 - 916*n^6 + 19950*n^5 - 241180*n^4 + 1748577*n^3 - 7604998*n^2 + 18375843*n - 19031658)*a(n-5) - (84*n^8 - 5096*n^7 + 135226*n^6 - 2050286*n^5 + 19428976*n^4 - 117838826*n^3 + 446719463*n^2 - 967742093*n + 917171710)*a(n-6) + (n-8)*(36*n^8 - 2284*n^7 + 63390*n^6 - 1005202*n^5 + 9960732*n^4 - 63153820*n^3 + 250166217*n^2 - 565970839*n + 559792740)*a(n-7) - (9*n^10 - 749*n^9 + 28038*n^8 - 621666*n^7 + 9040584*n^6 - 90096090*n^5 + 623077092*n^4 - 2952338109*n^3 + 9171809128*n^2 - 16867010733*n + 13941384550)*a(n-8) + (n-9)^10*(n-6)*a(n-9), for n>10. - _Vaclav Kotesovec_, Apr 25 2017
%F A285238 a(n) ~ n!*n/1024. - _Vaclav Kotesovec_, Apr 25 2017
%p A285238 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A285238       add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(
%p A285238       b(n-j, max(0, i-1)))*binomial(n-1, j-1)*
%p A285238       (j-1)!, j=1..n)))
%p A285238     end:
%p A285238 a:= n-> coeff(b(n, 10), x, 1):
%p A285238 seq(a(n), n=10..30);
%t A285238 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 A285238 a[n_] := Coefficient[b[n, 10], x, 1];
%t A285238 Table[a[n], {n, 10, 30}] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)
%Y A285238 Column k=10 of A185105.
%K A285238 nonn
%O A285238 10,2
%A A285238 _Alois P. Heinz_, Apr 15 2017