A293211 -id:A293211 - cp's OEIS frontend

A029575 Number of permutations of an n-set containing an 8-cycle.

0, 0, 0, 0, 0, 0, 0, 0, 5040, 45360, 453600, 4989600, 59875200, 778377600, 10897286400, 163459296000, 2451889440000, 41682120480000, 750278168640000, 14255285204160000, 285105704083200000, 5987219785747200000, 131718835286438400000, 3029533211588083200000

Offset: 0

Rob Pratt

nonn

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

Column k=8 of A293211.

a(n) = n! * (1 - sum(k=0, floor(n/8), (-1)^k/(k!*8^k) ) ); \\ Michel Marcus, Aug 08 2013

a(n) = n! * (1 - Sum_{k=0..floor(n/8)} (-1)^k/(k!*8^k) ).
a(n)/n! is asymptotic to 1-e^(-1/8).
E.g.f.: (1-exp(-x^k/k))/(1-x). - Alois P. Heinz, Oct 11 2017

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

Original entry on oeis.org

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

Rob Pratt

nonn

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

Column k=9 of A293211.

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

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

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)

A029577 Number of permutations of an n-set containing a 10-cycle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 362880, 3991680, 47900160, 622702080, 8717829120, 130767436800, 2092278988800, 35568742809600, 640237370572800, 12164510040883200, 231125690776780800, 4853639506312396800, 106780069138872729600, 2455941590194072780800

Offset: 0

Rob Pratt

nonn

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

Column k=10 of A293211.

a(n) = n! * (1 - sum(k=0, floor(n/10), (-1)^k/(k!*10^k) ) ); \\ Stéphane Rézel, Dec 11 2019

a(n) = n! * (1 - Sum_{k=0..floor(n/10)} (-1)^k/(k!10^k));
a(n)/n! is asymptotic to 1-e^(-1/10).
E.g.f.: (1-exp(-x^10/10))/(1-x). - Alois P. Heinz, Oct 11 2017
Conjectures from Stéphane Rézel, Dec 11 2019: (Start)
Recurrence: a(n) = n*a(n-1), for n > 10 and n !== 0 (mod 10);
for k > 1, a(10*k) = a(10*k-1)*S(k)/S(k-1) where S(k) = 10*k*S(k-1) - (-1)^k with S(1) = 1.
(End)

cp's OEIS Frontend

A029575 Number of permutations of an n-set containing an 8-cycle.

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

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Mathematica

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A029577 Number of permutations of an n-set containing a 10-cycle.

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