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%I #46 Feb 23 2022 06:36:16
%S 1,1,6,69,1336,39145,1598256,85996561,5872177536,494848403793,
%T 50333180780800,6068500612311841,854434117410352128,
%U 138752719761249646585,25714777079368557164544,5389541081414619785888625,1267387594395443339970052096,332074775201035547446532113825
%N Number of ordered pairs of endofunctions (f,g) on a set of n elements satisfying f(x) = g(f(f(x))).
%C This also counts pairs (f,g) satisfying f(x) = g(f^{r}(x)) for r > 1. - _David Einstein_, Nov 18 2016
%H Alois P. Heinz, <a href="/A235328/b235328.txt">Table of n, a(n) for n = 0..263</a>
%F a(n) = Sum_{k=1..n} k! * C(n, k) * (n*k)^(n-k). - _David Einstein_, Oct 10 2016
%F a(n) = n! * [x^n] 1/(1 - x*exp(n*x)). - _Ilya Gutkovskiy_, Nov 26 2017
%F log(a(n)) ~ log(sqrt(2*Pi) * n^(2*n - n/LambertW(exp(1)*n) + 1/2) / (LambertW(exp(1)*n) * exp(n/LambertW(exp(1)*n)) * (LambertW(exp(1)*n) - 1)^(n*(1 - 1/LambertW(exp(1)*n))))). - _Vaclav Kotesovec_, Feb 20 2022
%F More precise asymptotics: a(n) ~ sqrt(2*Pi) * (w^2 - w - 1 + 2/w) * exp(n*(1/w^3 - 1/w)) * n^(2*n + n/w^3 - n/w + 1/2) * (w^2 - 1)^(n*(1 + 1/w^3 - 1/w)) * (1 - w^2 + w^3)^(n/w - n - n/w^3 - 1), where w = LambertW(exp(1)*n). - _Vaclav Kotesovec_, Feb 23 2022
%p a:= proc(n) option remember; local b; b:=
%p proc(m, i) option remember; `if`(m=0, n^i, `if`(i<1, 0,
%p add(b(m-j, i-1)*binomial(m, j)*j, j=0..m)))
%p end: forget(b):
%p b(n$2)
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jul 23 2014
%t a[n_] := If[n==0, 1, Sum[k! Binomial[n, k] (n k)^(n - k), {k, 1, n}]]
%t Table[a[n],{n,20}] (* _David Einstein_, Oct 10 2016 *)
%Y Cf. A000248, A000949, A181162, A351795.
%K nonn
%O 0,3
%A _Chad Brewbaker_, Mar 26 2014
%E a(6)-a(7) from _Giovanni Resta_, Mar 26 2014
%E a(8)-a(17) from _Alois P. Heinz_, Jul 23 2014
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