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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.

%I A222379 #36 Mar 05 2019 18:04:50
%S A222379 0,1,0,1,1,4,6,19,38,107,247,668,1666,4468,11603,31210,83044,224893,
%T A222379 607658,1657966,4528193,12441364,34254321,94696165,262389581,
%U A222379 729258392,2031264865,5671570468,15867219821,44480785907,124913622052,351393746745,990048748684
%N 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.
%C A222379 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 A222379 Alois P. Heinz, <a href="/A222379/b222379.txt">Table of n, a(n) for n = 0..1000</a>
%H A222379 Alois P. Heinz, <a href="/A222379/a222379.jpg">Plot of A000081(8) = 115 = 77 + 38 functions with 8 x's</a>
%H A222379 Wikipedia, <a href="https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero">Zero to the power of zero</a>
%F A222379 A222380(n) + a(n) = A000081(n).
%F A222379 A222380(n) - a(n) = A211192(n).
%F A222379 a(n) = Sum_{i=A087803(n-1)+1..A087803(n)} (1-A306710(i)).
%e A222379 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.
%p A222379 g:= proc(n, i) option remember; `if`(n=0, [0, 1], `if`(i<1, 0, (v->[v[1]-
%p A222379       v[2], v[2]])(add(((l, h)-> [binomial(l[2]+l[1]+j-1, j)*(h[1]+h[2]),
%p A222379       binomial(l[1]+j-1, j)*h[2]])(g(i-1$2), g(n-i*j, i-1)), j=0..n/i))))
%p A222379     end:
%p A222379 a:= n-> g(n-1$2)[2]:
%p A222379 seq(a(n), n=0..40);
%t A222379 f[l_, h_] := {Binomial[l[[2]] + l[[1]] + j - 1, j]*(h[[1]] + h[[2]]), Binomial[l[[1]] + j - 1, j]*h[[2]]};
%t A222379 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]}]]]];
%t A222379 a[n_] := g[n - 1, n - 1][[2]];
%t A222379 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 27 2019, after _Alois P. Heinz_ *)
%Y A222379 Cf. A000081, A000108, A055113, A087803, A211192, A215703, A222380, A306710.
%K A222379 nonn
%O A222379 0,6
%A A222379 _Alois P. Heinz_, Feb 17 2013