I am interested in the following sequence: $$ T_n = \sum\limits^{n-1}_{k=0} \begin{pmatrix} n \\ k \end{pmatrix} T_{k}, \ \ \ \ T_0 = C \in \mathbb{N} $$ I would like to express it as a function of n, but none of the method I have tried work.

Asymptotically, I can tell that $T_n = \mathcal{O}(2^{\frac{k^2}{2}})$. One method that failed was to see $T_n$ as the $n$-th term in a series, but those terms grow to fast for it to work.

Do you know how to solve it, or have an intuition regarding how it might get solved?

Thank you.

`sage: T = lambda n: x if n == 0 else sum(binomial(n,k)*T(k) for k in range(n))`

`sage: T = cached_function(T)`

`sage: [T(n) for n in range(5)]`

`[x, x, 3*x, 13*x, 75*x]`

`sage: oeis([T(n)/x for n in range(20)])`

`0: A000670: Fubini numbers: number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements; or number of ordered partitions of [n].`

$\endgroup$ – Martin Rubey Feb 18 at 12:53