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A285237
Number of entries in the ninth cycles of all permutations of [n].
2
1, 47, 1434, 36792, 872511, 20014299, 455265257, 10420963144, 242208466145, 5748862140283, 139849088103596, 3494752531722564, 89838192687840304, 2377612074981717632, 64807344109730799968, 1819505580964336136560, 52611858820598185363536
OFFSET
9,2
COMMENTS
Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
LINKS
Wikipedia, Permutation
FORMULA
a(n) = A185105(n,9).
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
a(n) ~ n!*n/512. - Vaclav Kotesovec, Apr 25 2017
MAPLE
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(
b(n-j, max(0, i-1)))*binomial(n-1, j-1)*
(j-1)!, j=1..n)))
end:
a:= n-> coeff(b(n, 9), x, 1):
seq(a(n), n=9..30);
MATHEMATICA
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}]]];
a[n_] := Coefficient[b[n, 9], x, 1];
Table[a[n], {n, 9, 30}] (* Jean-François Alcover, Jun 01 2018, from Maple *)
CROSSREFS
Column k=9 of A185105.
Sequence in context: A047911 A009069 A348805 * A157359 A153214 A142845
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 15 2017
STATUS
approved