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A285237 Number of entries in the ninth cycles of all permutations of [n].

%I A285237 #13 Jun 01 2018 08:10:37
%S A285237 1,47,1434,36792,872511,20014299,455265257,10420963144,242208466145,
%T A285237 5748862140283,139849088103596,3494752531722564,89838192687840304,
%U A285237 2377612074981717632,64807344109730799968,1819505580964336136560,52611858820598185363536
%N A285237 Number of entries in the ninth cycles of all permutations of [n].
%C A285237 Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%H A285237 Alois P. Heinz, <a href="/A285237/b285237.txt">Table of n, a(n) for n = 9..450</a>
%H A285237 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A285237 a(n) = A185105(n,9).
%F A285237 Recurrence: (n-9)*(n-6)*a(n) = 2*(4*n^3 - 77*n^2 + 471*n - 916)*a(n-1) - 14*(2*n^4 - 49*n^3 + 439*n^2 - 1714*n + 2474)*a(n-2) + 14*(n-5)*(4*n^4 - 103*n^3 + 981*n^2 - 4117*n + 6454)*a(n-3) - 7*(n-6)*(10*n^5 - 320*n^4 + 4070*n^3 - 25770*n^2 + 81333*n - 102427)*a(n-4) + 14*(4*n^7 - 185*n^6 + 3661*n^5 - 40195*n^4 + 264477*n^3 - 1042986*n^2 + 2282488*n - 2138058)*a(n-5) - (28*n^8 - 1554*n^7 + 37702*n^6 - 522242*n^5 + 4517128*n^4 - 24979724*n^3 + 86233855*n^2 - 169871843*n + 146155098)*a(n-6) + (8*n^9 - 526*n^8 + 15356*n^7 - 261226*n^6 + 2853242*n^5 - 20747608*n^4 + 100420076*n^3 - 311890495*n^2 + 563892963*n - 452026202)*a(n-7) - (n-8)^9*(n-5)*a(n-8), for n>9. - _Vaclav Kotesovec_, Apr 25 2017
%F A285237 a(n) ~ n!*n/512. - _Vaclav Kotesovec_, Apr 25 2017
%p A285237 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A285237       add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(
%p A285237       b(n-j, max(0, i-1)))*binomial(n-1, j-1)*
%p A285237       (j-1)!, j=1..n)))
%p A285237     end:
%p A285237 a:= n-> coeff(b(n, 9), x, 1):
%p A285237 seq(a(n), n=9..30);
%t A285237 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 A285237 a[n_] := Coefficient[b[n, 9], x, 1];
%t A285237 Table[a[n], {n, 9, 30}] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)
%Y A285237 Column k=9 of A185105.
%K A285237 nonn
%O A285237 9,2
%A A285237 _Alois P. Heinz_, Apr 15 2017