|
%I #11 Sep 23 2022 18:45:23
%S 1,0,0,0,0,0,0,0,0,0,362880,39916800,2874009600,175394419200,
%T 9967384627200,551675124000000,30471021291110400,1703458301210265600,
%U 97213825272736972800,5693251850259515942400,343266731083210449715200,21349233350716392722764800
%N Number of endofunctions on [n] whose cycle lengths are multiples of 10.
%C 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
%H Alois P. Heinz, <a href="/A246617/b246617.txt">Table of n, a(n) for n = 0..300</a>
%F E.g.f.: 1/(1-LambertW(-x)^10)^(1/10). - _Vaclav Kotesovec_, Sep 01 2014
%F 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
%p with(combinat):
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i>n, 0, add(b(n-i*j, i+10)*(i-1)!^j*
%p multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
%p end:
%p a:= a->add(b(j, 10)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p seq(a(n), n=0..25);
%t CoefficientList[Series[1/(1-LambertW[-x]^10)^(1/10),{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Sep 01 2014 *)
%Y Column k=10 of A246609.
%K nonn
%O 0,11
%A _Alois P. Heinz_, Aug 31 2014
|
