OFFSET
0,5
COMMENTS
The total number of binary bracketings of 0^0^...^0 with n 0's is A000108(n-1) for n > 0.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1670
Eric Weisstein's World of Mathematics, Binary Bracketing
Wikipedia, Zero to the power of zero
EXAMPLE
There are A000108(3) = 5 binary bracketings of 0^0^0^0: ((0^0)^0)^0, (0^0)^(0^0), (0^(0^0))^0, 0^((0^0)^0), 0^(0^(0^0)). Only 0^((0^0)^0) evaluates to 0: 0^((0^0)^0) = 0^(1^0) = 0^1 = 0. The four other bracketings evaluate to 1. Thus a(4) = 4-1 = 3.
MAPLE
b:= proc(n) option remember; `if`(n<2, [n, 0], add(((f, g)-> [f[1]*g[2],
f[1]*g[1] +f[2]*g[1] +f[2]*g[2]])(b(i), b(n-i)), i=1..n-1))
end:
a:= n-> (v-> v[2]-v[1])(b(n)):
seq(a(n), n=0..29);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, -n, ((35*n^3-147*n^2+220*n-120)*
a(n-1)+18*(n-2)*(5*n-6)*(2*n-5)*a(n-2))/((2*(5*n-11))*(2*n-1)*n))
end:
seq(a(n), n=0..29);
MATHEMATICA
a[n_] := a[n] = If[n<2, -n, ((35n^3 - 147n^2 + 220n - 120) a[n-1] + 18(n-2) (5n - 6)(2n - 5) a[n-2])/((2(5n - 11))(2n - 1)n)];
a /@ Range[0, 29] (* Jean-François Alcover, Apr 02 2021, after 2nd Maple program *)
CROSSREFS
KEYWORD
sign
AUTHOR
Alois P. Heinz, Mar 04 2019
STATUS
approved