A306710 - OEIS

%I #23 Mar 06 2019 15:17:57

%S 0,1,1,0,1,1,0,1,1,1,0,1,1,0,0,1,0,1,1,1,1,1,0,1,1,1,1,1,0,0,1,0,0,1,

%T 1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1,1,1,0,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,

%U 0,0,1,0,0,0,0,0,1,1,0,1,1,0,0,1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1

%N a(n) = f_n(0), where f_n is the n-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways, with conventions that 0^0=1^0=1^1=1, 0^1=0.

%C The ordering of the functions f_n is defined in A215703: f_1, f_2, ... = x, x^x, x^(x^2), x^(x^x), x^(x^3), x^(x^x*x), x^(x^(x^2)), x^(x^(x^x)), x^(x^4), x^(x^x*x^2), ... .

%H Alois P. Heinz, <a href="/A306710/b306710.txt">Table of n, a(n) for n = 1..20299</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero">Zero to the power of zero</a>

%F Sum_{i=A087803(n-1)+1..A087803(n)}    a(i)  = A222380(n).

%F Sum_{i=A087803(n-1)+1..A087803(n)} (1-a(i)) = A222379(n).

%e a(1) = f_1(0) = x_{x=0} = 0.

%e a(2) = f_2(0) = x^x_{x=0} = 0^0 = 1.

%e a(3) = f_3(0) = x^(x^2)_{x=0} = 0^(0^2) = 0^0 = 1.

%e a(4) = f_4(0) = x^(x^x)_{x=0} = 0^(0^0) = 0^1 = 0.

%p T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:

%p g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(

%p       seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=

%p       combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])

%p     end:

%p a:= proc() local i, l; i, l:= 0, []; proc(n) while n>nops(l)

%p       do i:= i+1; l:= [l[], subs(x=0, T(i))[]] od; l[n] end

%p     end():

%p seq(a(n), n=1..120);

%Y Cf. A000081, A087803, A215703, A222379, A222380.

%Y Partial sums give A306726.

%K nonn

%O 1

%A _Alois P. Heinz_, Mar 05 2019