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%I #79 Mar 04 2019 12:18:13
%S 0,-1,1,0,2,1,8,10,39,72,225,506,1434,3550,9767,25391,69293,185061,
%T 505843,1372744,3769842,10339104,28546539,78890525,218945822,
%U 608657861,1697106780,4740593393,13272626627,37224982494,104599603493,294384019508,829836855332
%N Consider all distinct functions f representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways; sequence gives difference between numbers of f with f(0)=1 and numbers of f with f(0)=0, with conventions that 0^0=1^0=1^1=1, 0^1=0.
%C A000081(n) distinct functions are representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways. Some functions are representable in more than one way, the number of valid parenthesizations is A000108(n-1) for n>0.
%H Alois P. Heinz, <a href="/A211192/b211192.txt">Table of n, a(n) for n = 0..1000</a>
%H Alois P. Heinz, <a href="/A222379/a222379.jpg">Plot of A000081(8) = 115 = 77 + 38 functions with 8 x's</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero">Zero to the power of zero</a>
%F a(n) = A222380(n) - A222379(n).
%F From _Alois P. Heinz_, Mar 01 2019: (Start)
%F a(n) is even <=> n in { A258592 }.
%F a(n) is odd <=> n in { A263831 }. (End)
%e There are A000081(4) = 4 functions f representable as x -> x^x^...^x with 4 x's and parentheses inserted in all possible ways: ((x^x)^x)^x, (x^x)^(x^x) == (x^(x^x))^x, x^((x^x)^x), x^(x^(x^x)). Only x^((x^x)^x) evaluates to 0 at x=0: 0^((0^0)^0) = 0^(1^0) = 0^1 = 0. Three functions evaluate to 1 at x=0: ((0^0)^0)^0 = (1^0)^0 = 1^0 = 1, (0^0)^(0^0) = 1^1 = 1, 0^(0^(0^0)) = 0^(0^1) = 0^0 = 1. Thus a(4) = 3-1 = 2.
%e a(8) = A222380(8) - A222379(8) = 77 - 38 = 39.
%p g:= proc(n, i) option remember; `if`(n=0, [0, 1], `if`(i<1, 0, (v->[v[1]-
%p v[2], v[2]])(add(((l, h)-> [binomial(l[2]+l[1]+j-1, j)*(h[1]+h[2]),
%p binomial(l[1]+j-1, j)*h[2]])(g(i-1$2), g(n-i*j, i-1)), j=0..n/i))))
%p end:
%p a:= n-> (f-> f[1]-f[2])(g(n-1$2)):
%p seq(a(n), n=0..40);
%t g[n_, i_] := g[n, i] = If[n==0, {0, 1}, If[i<1, {0, 0}, ({#[[1]]-#[[2]], #[[2]]}&)[Sum[Function[{l, h}, {(h[[1]]+h[[2]])*Binomial[j+l[[1]]+l[[2]] -1, j], h[[2]]*Binomial[j+l[[1]]-1, j]}][g[i-1, i-1]], g[n-i*j, i-1]]], {j, 0, Quotient[n, i]}]];
%t a[n_] := (#[[1]]-#[[2]]&)[g[n-1, n-1]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 22 2017, translated from Maple *)
%Y Cf. A000081, A000108, A055113, A215703, A222379, A222380, A258592, A263831, A306668.
%K sign
%O 0,5
%A _Alois P. Heinz_, Feb 18 2013
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