A222379 Number of distinct functions f representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways giving result f(0)=0, with conventions that 0^0=1^0=1^1=1, 0^1=0.

0, 1, 0, 1, 1, 4, 6, 19, 38, 107, 247, 668, 1666, 4468, 11603, 31210, 83044, 224893, 607658, 1657966, 4528193, 12441364, 34254321, 94696165, 262389581, 729258392, 2031264865, 5671570468, 15867219821, 44480785907, 124913622052, 351393746745, 990048748684

Offset: 0

Author

Alois P. Heinz, Feb 17 2013

Keywords

• nonn

Comments

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.

Examples

			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. Thus a(4) = 1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Alois P. Heinz, Plot of A000081(8) = 115 = 77 + 38 functions with 8 x's
• Wikipedia, Zero to the power of zero

Crossrefs

Cf. A000081, A000108, A055113, A087803, A211192, A215703, A222380, A306710.

Programs

Maple

g:= proc(n, i) option remember; `if`(n=0, [0, 1], `if`(i<1, 0, (v->[v[1]-
      v[2], v[2]])(add(((l, h)-> [binomial(l[2]+l[1]+j-1, j)*(h[1]+h[2]),
      binomial(l[1]+j-1, j)*h[2]])(g(i-1$2), g(n-i*j, i-1)), j=0..n/i))))
    end:
a:= n-> g(n-1$2)[2]:
seq(a(n), n=0..40);

Mathematica

f[l_, h_] := {Binomial[l[[2]] + l[[1]] + j - 1, j]*(h[[1]] + h[[2]]), Binomial[l[[1]] + j - 1, j]*h[[2]]};
g[n_, i_] := g[n, i] = If[n == 0, {0, 1}, If[i < 1, {0, 0}, Function[v, {v[[1]] - v[[2]], v[[2]]}][Sum[f[g[i - 1, i - 1], g[n - i*j, i - 1]], {j, 0, Quotient[n, i]}]]]];
a[n_] := g[n - 1, n - 1][[2]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 27 2019, after Alois P. Heinz *)

Formula

A222380(n) + a(n) = A000081(n).

A222380(n) - a(n) = A211192(n).

a(n) = Sum_{i=A087803(n-1)+1..A087803(n)} (1-A306710(i)).