A246617 Number of endofunctions on [n] whose cycle lengths are multiples of 10.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 362880, 39916800, 2874009600, 175394419200, 9967384627200, 551675124000000, 30471021291110400, 1703458301210265600, 97213825272736972800, 5693251850259515942400, 343266731083210449715200, 21349233350716392722764800

Offset: 0

Author

Alois P. Heinz, Aug 31 2014

Keywords

• nonn

Comments

In general, column k of A246609 is (for k > 1) asymptotic to n^(n-1/2 + 1/(2*k)) * sqrt(2*Pi) / (2^(1/(2*k)) * k^(1/k) * Gamma(1/(2*k))) * (1 - (3*k-1)*(k-1) * sqrt(2/n) * Gamma(1/(2*k)) / (12 * k^2 * Gamma(1/2 + 1/(2*k)))). - Vaclav Kotesovec, Sep 01 2014

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Column k=10 of A246609.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+10)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 10)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

Mathematica

CoefficientList[Series[1/(1-LambertW[-x]^10)^(1/10),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

Formula

E.g.f.: 1/(1-LambertW(-x)^10)^(1/10). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-9/20) * 2^(7/20) * sqrt(Pi) / (5^(1/10) * Gamma(1/20)) * (1 - 87 * sqrt(2/n) * Gamma(1/20) / (400 * Gamma(11/20))). - Vaclav Kotesovec, Sep 01 2014