The square root of the square root of the square root of the…
APL (Dyalog Unicode), 39 bytes

+/⊢{∨/⍺⍵&lt;⍵0:0⋄⍺=0:1⋄+/∊∇¨/⍺(⍵*2)-⊂⍳⍺}¨⍳

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A tacit function containing an inner dfn to use recursion. Does not use floating point numbers at all.
How it works
First of all, observe that
$$
\displaystyle
\sqrt{a_1+\sqrt{a_2 + \cdots + \stackrel{\vdots}{\sqrt{a_t}}}} \le \cdots \le \sqrt{a_1+a_2 + \cdots + a_t} \le a_1+a_2 + \cdots + a_t = n
$$
and this holds for all suffixes of any given sequence of positive integers.
Let's define a function \$f(x,y)\$ as the number of sequences where the sum is \$x\$ and the &quot;root sum&quot; is \$y\$. Then the following holds:
$$
\begin{align}
f(0, 0) &amp;= 1 \\
f(0, y) &amp;= 0, \qquad 0 &lt; y \\
f(x, y) &amp;= 0, \qquad x &lt; y \text{ or } y &lt; 0 \\
f(x, y) &amp;= \sum_{i=1}^{x}{f(x-i, y^2-i)}
\end{align}
$$
Then the desired result is the sum \$\sum_{i=1}^{n}{f(n,i)}\$.