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

0, 0, 0, 0, 0, 0, 0, 0, 0, 40320, 403200, 4435200, 53222400, 691891200, 9686476800, 145297152000, 2324754432000, 39520825344000, 671854030848000, 12765226586112000, 255304531722240000, 5361395166167040000, 117950693655674880000, 2712865954080522240000

Offset: 0

Author

Rob Pratt

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Column k=9 of A293211.

Programs

Mathematica

Table[n!*(1 - Sum[(-1)^k/(k!*9^k), {k, 0, Floor[n/9]}]), {n, 0, 25}] (* Paolo Xausa, Aug 11 2024 *)

PARI

a(n) = n! * (1 - sum(k=0, floor(n/9), (-1)^k/(k!*9^k) ) ); \\ Stéphane Rézel, Nov 24 2019

Formula

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).

E.g.f.: (1-exp(-x^9/9))/(1-x). - Alois P. Heinz, Oct 11 2017

Conjectures from Stéphane Rézel, Nov 24 2019: (Start)

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

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.

(End)