A306710 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.

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, 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, 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

Offset

1

Comments

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), ... .

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20299

Wikipedia, Zero to the power of zero

Formula

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

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

Example

a(1) = f_1(0) = x_{x=0} = 0.
a(2) = f_2(0) = x^x_{x=0} = 0^0 = 1.
a(3) = f_3(0) = x^(x^2)_{x=0} = 0^(0^2) = 0^0 = 1.
a(4) = f_4(0) = x^(x^x)_{x=0} = 0^(0^0) = 0^1 = 0.

Maple

T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
      seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
      combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
    end:
a:= proc() local i, l; i, l:= 0, []; proc(n) while n>nops(l)
      do i:= i+1; l:= [l[], subs(x=0, T(i))[]] od; l[n] end
    end():
seq(a(n), n=1..120);

Crossrefs

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

Partial sums give A306726.

Sequence in context: A000494 A022933 A194683 * A188295 A228039 A163532

Adjacent sequences: A306707 A306708 A306709 * A306711 A306712 A306713

Keyword

nonn

Author

Alois P. Heinz, Mar 05 2019

Status

approved