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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.
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%I #16 Mar 11 2019 10:08:08

%S 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 20,20,21,21,21,22,23,23,24,25,26,27,27,28,29,30,31,31,32,33,34,34,35,

%U 35,36,37,38,38,39,40,41,42,43,44,45,46,47,47,47,47,47

%N 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 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 Alois P. Heinz, <a href="/A306726/b306726.txt">Table of n, a(n) for n = 1..53272</a>

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

%F a(n) = Sum_{i=1..n} A306710(i).

%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 b:= 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 a:= proc(n) option remember; `if`(n<1, 0, b(n)+a(n-1)) end:

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

%Y Partial sums of A306710.

%Y Cf. A000081, A087803, A215703.

%K nonn

%O 1,3

%A _Alois P. Heinz_, Mar 06 2019