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A235328
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Number of ordered pairs of endofunctions (f,g) on a set of n elements satisfying f(x) = g(f(f(x))).
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8
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1, 1, 6, 69, 1336, 39145, 1598256, 85996561, 5872177536, 494848403793, 50333180780800, 6068500612311841, 854434117410352128, 138752719761249646585, 25714777079368557164544, 5389541081414619785888625, 1267387594395443339970052096, 332074775201035547446532113825
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OFFSET
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0,3
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COMMENTS
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This also counts pairs (f,g) satisfying f(x) = g(f^{r}(x)) for r > 1. - David Einstein, Nov 18 2016
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n} k! * C(n, k) * (n*k)^(n-k). - David Einstein, Oct 10 2016
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
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
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MAPLE
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a:= proc(n) option remember; local b; b:=
proc(m, i) option remember; `if`(m=0, n^i, `if`(i<1, 0,
add(b(m-j, i-1)*binomial(m, j)*j, j=0..m)))
end: forget(b):
b(n$2)
end:
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MATHEMATICA
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a[n_] := If[n==0, 1, Sum[k! Binomial[n, k] (n k)^(n - k), {k, 1, n}]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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