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.

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, 20, 20, 21, 21, 21, 22, 23, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 31, 32, 33, 34, 34, 35, 35, 36, 37, 38, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 47, 47, 47, 47

Offset: 1

Author

Alois P. Heinz, Mar 06 2019

Keywords

• nonn

Comments

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), ... .

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..53272
• Alois P. Heinz, Animation of scaled plots [n/j-> a(n)/a(j)] for n = 1..j, and j = A087803(2)..A087803(15)
• Wikipedia, Zero to the power of zero

Crossrefs

Partial sums of A306710.

Cf. A000081, A087803, A215703.

Programs

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:
b:= 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():
a:= proc(n) option remember; `if`(n<1, 0, b(n)+a(n-1)) end:
seq(a(n), n=1..120);

Formula

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