For simplicity, I assume that $t$ is fixed and omit it as the second index in $T$.

First, let's focus on initial terms with $n\leq t$, for which
$$T_n = \sum_{k=0}^{n-1} T_k T_{(t-1-k)\bmod n},$$
implying in particular that $T_1 = 1$ and 
$$T_t = \sum_{k=0}^{t-1} T_k T_{t-1-k}.$$

Let 
$$p(x) := \sum_{n=0}^{t-1} T_n x^n$$
be the generating polynomial for the initial terms.

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For $n\geq t$, we have
\begin{split}
T_n &= \sum_{k=0}^{t-1} T_k T_{t-1-k} + \sum_{k=t}^{n-1} T_k T_{n-1+t-k} \\
& = T_t + \sum_{k=0}^{n+t-1} T_k T_{n-1+t-k} - \sum_{k=0}^{t-1} T_k T_{n-1+t-k} - \sum_{k=n}^{n+t-1} T_k T_{n-1+t-k}\\
& = T_t + \sum_{k=0}^{n+t-1} T_k T_{n-1+t-k} - 2\sum_{k=0}^{t-1} T_k T_{n-1+t-k}\\
\end{split}

Let $F(x):=\sum_{n\geq 0} T_n x^n$ be the generating function, and $q(x)$ be the polynomial formed by the initial terms of degree $<2t-1$ from $F(x)^2$ and $r(x)$ be a similar polynomial from $F(x)p(x)$. 

Then the recurrence for $T_n$ with $n\geq t$ translates into
$$x^{t-1}(F(x) - p(x)) = T_t\frac{x^{2t-1}}{1-x} + (F(x)^2 - q(x)) - 2(F(x)p(x)-r(x)),$$
which is a quadratic equation for $F(x)$. Solving it, we get an explicit expression for the generating function:
$$F(x) = p(x)+\frac{1}{2}x^{t-1} - \sqrt{p(x)^2+\frac{1}{4}x^{2t-2}+q(x)-2r(x)-T_t\frac{x^{2t-1}}{1-x}}.$$

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**Example for $t=1$.** In this case, $T_t=1$, $p(x)=q(x)=r(x)=1$, implying that
$$F(x) = \frac12\big(3-\sqrt{\frac{1-5x}{1-x}}\big).$$

**Example for $t=2$.** In this case, $T_t=2$, $p(x)=1+x$, $q(x)=1+2x+5x^2$, and $r(x)=1+2x+3x^2$, implying that
$$F(x) = 1 + \frac{x}{2}\big( 3 - \sqrt{\frac{1-9x}{1-x}}\big).$$

**Example for $t=3$.** In this case, $T_t=5$, $p(x)=1+x+2x^2$, $q(x)=1 + 2x + 5x^2 + 14x^3 + 74x^4$, and $r(x)=1 + 2x + 5x^2 + 9x^3 + 39x^4$, implying that
$$F(x) = 1 + x + x^2\big(5-\sqrt{\frac{1-21x}{1-x}}\big).$$

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The examples suggest

**Conjecture.** For any $t$, the generating function for $T_k$ has the form:
$$F(x) = p(x) + \frac{x^{t-1}}{2}\big(1 - \sqrt{\frac{1-(4T_t+1)x}{1-x}}\big).$$

The sequence of $T_t$ for $t=1,2,\dots$ starts with
$$1, 2, 5, 14, 50, 194, 1319, 8834, 117869, 1269734, 560616089, \dots$$
giving the corresponding sequence of coefficients $4T_t+1$:
$$5, 9, 21, 57, 201, 777, 5277, 35337, 471477, 5078937, 2242464357, \dots$$