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A029576 Number of permutations of an n-set containing a 9-cycle. 2

%I

%S 0,0,0,0,0,0,0,0,0,40320,403200,4435200,53222400,691891200,9686476800,

%T 145297152000,2324754432000,39520825344000,671854030848000,

%U 12765226586112000,255304531722240000,5361395166167040000,117950693655674880000,2712865954080522240000

%N Number of permutations of an n-set containing a 9-cycle.

%H Alois P. Heinz, <a href="/A029576/b029576.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) = n!*(1 - Sum_{k=0..floor(n/9)} (-1)^k/(k!*9^k); a(n)/n! is asymptotic to 1-e^(-1/9).

%F E.g.f.: (1-exp(-x^9/9))/(1-x). - _Alois P. Heinz_, Oct 11 2017

%F Conjectures from _Stéphane Rézel_, Nov 24 2019: (Start)

%F Recurrence: a(n) = n*a(n-1), for n > 9 and n !== 0 (mod 9);

%F for k > 1, a(9*k) = a(9*k-1)*S(k)/S(k-1) where S(k) = 9*k*S(k-1) - (-1)^k with S(1) = 1.

%F (End)

%o (PARI) a(n) = n! * (1 - sum(k=0, floor(n/9), (-1)^k/(k!*9^k) ) ); \\ _Stéphane Rézel_, Nov 24 2019

%Y Column k=9 of A293211.

%K nonn

%O 0,10

%A _Rob Pratt_

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Last modified October 17 10:58 EDT 2021. Contains 348048 sequences. (Running on oeis4.)