cp's OEIS Frontend

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

%I A029576 #26 Aug 11 2024 22:06:25
%S A029576 0,0,0,0,0,0,0,0,0,40320,403200,4435200,53222400,691891200,9686476800,
%T A029576 145297152000,2324754432000,39520825344000,671854030848000,
%U A029576 12765226586112000,255304531722240000,5361395166167040000,117950693655674880000,2712865954080522240000
%N A029576 Number of permutations of an n-set containing a 9-cycle.
%H A029576 Alois P. Heinz, <a href="/A029576/b029576.txt">Table of n, a(n) for n = 0..450</a>
%F A029576 a(n) = n!*(1 - Sum_{k=0..floor(n/9)} (-1)^k/(k!*9^k)).
%F A029576 a(n)/n! is asymptotic to 1-e^(-1/9).
%F A029576 E.g.f.: (1-exp(-x^9/9))/(1-x). - _Alois P. Heinz_, Oct 11 2017
%F A029576 Conjectures from _Stéphane Rézel_, Nov 24 2019: (Start)
%F A029576 Recurrence: a(n) = n*a(n-1), for n > 9 and n !== 0 (mod 9);
%F A029576 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 A029576 (End)
%t A029576 Table[n!*(1 - Sum[(-1)^k/(k!*9^k), {k, 0, Floor[n/9]}]), {n, 0, 25}] (* _Paolo Xausa_, Aug 11 2024 *)
%o A029576 (PARI) a(n) = n! * (1 - sum(k=0, floor(n/9), (-1)^k/(k!*9^k) ) ); \\ _Stéphane Rézel_, Nov 24 2019
%Y A029576 Column k=9 of A293211.
%K A029576 nonn
%O A029576 0,10
%A A029576 _Rob Pratt_