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A306726 a(n) = Sum_{i=1..n} f_i(0), where f_i is the i-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.

%I A306726 #16 Mar 11 2019 10:08:08
%S A306726 0,1,2,2,3,4,4,5,6,7,7,8,9,9,9,10,10,11,12,13,14,14,15,16,17,18,19,20,
%T A306726 20,20,21,21,21,22,23,23,24,25,26,27,27,28,29,30,31,31,32,33,34,34,35,
%U A306726 35,36,37,38,38,39,40,41,42,43,44,45,46,47,47,47,47,47
%N A306726 a(n) = Sum_{i=1..n} f_i(0), where f_i is the i-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 A306726 The ordering of the functions f_i 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 A306726 Alois P. Heinz, <a href="/A306726/b306726.txt">Table of n, a(n) for n = 1..53272</a>
%H A306726 Alois P. Heinz, <a href="/A306726/a306726.gif">Animation of scaled plots [n/j-> a(n)/a(j)] for n = 1..j, and j = A087803(2)..A087803(15)</a>
%H A306726 Wikipedia, <a href="https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero">Zero to the power of zero</a>
%F A306726 a(n) = Sum_{i=1..n} A306710(i).
%p A306726 T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
%p A306726 g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
%p A306726       seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
%p A306726       combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
%p A306726     end:
%p A306726 b:= proc() local i, l; i, l:= 0, []; proc(n) while n>nops(l)
%p A306726       do i:= i+1; l:= [l[], subs(x=0, T(i))[]] od; l[n] end
%p A306726     end():
%p A306726 a:= proc(n) option remember; `if`(n<1, 0, b(n)+a(n-1)) end:
%p A306726 seq(a(n), n=1..120);
%Y A306726 Partial sums of A306710.
%Y A306726 Cf. A000081, A087803, A215703.
%K A306726 nonn
%O A306726 1,3
%A A306726 _Alois P. Heinz_, Mar 06 2019