OFFSET
1,5
COMMENTS
A000081(n) distinct functions are representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways. The number of valid parenthesizations is A000108(n-1). So the number of redundant representations is A000108(n-1) - A000081(n).
For n>=6 we have a(n) > A000081(n), so the number of redundant function representations is larger than the number of essential representations.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
EXAMPLE
MAPLE
with(numtheory):
b:= proc(n) option remember; `if`(n<=1, n,
(add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
end:
C:= n-> binomial(2*n, n)/(n+1):
a:= n-> C(n-1) -b(n):
seq(a(n), n=1..40);
MATHEMATICA
b[n_] := b[n] = If[n <= 1, n, Sum[DivisorSum[j, #*b[#]&]*b[n-j], {j, 1, n-1}]/(n-1)];
c[n_] := Binomial[2*n, n]/(n+1);
a[n_] := c[n-1] - b[n];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 30 2012
STATUS
approved