Consider
$$ a(1) = 1 $$
$$ a(2) = a(1) a(1) = 1 $$
$$ a(3) = a(1) a(2) + 2 a(1) a(2) = 3 $$
$$ a(4) = a(1) a(3) + 2 a(2) a(2) + 3 a(3) a(1) = 14 $$
$$ a(5) = a(1) a(4) + 2 a(2) a(3) + 3 a(3) a(2) + 4 a(4) a(1) = 14 + 6 + 9 + 56 = 85 $$
$$ a(n) = 1 a(1) a(n-1) + 2 a(2) a(n-2) + 3 a(3) a(n-3) + ... $$
This recursion reminds me of Catalan numbers. Also factorial numbers and fibonacci(2 n) have similar recursions.
Im not sure if this is “ combinatorics “ or “ calculus “ like the Catalan Numbers.
I wonder If there is a closed form. It seems to grow like $ \frac{n^2 n!}{5} $ but that is a very brute estimate.
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Slightly more general
$$ a(n,K) = \sum_{i = 1}^{n-1} i^K a(i,K) a(n-i,K) $$
for Nonzero K is a mystery to me.
Closed form ? Accurate growth rate ( asymptotic ) ??
Is this a known generalization of Catalan Numbers ??
For $$a_n=\sum_{i=1}^{n-1} i\, a_i\, a_{n-1} \qquad \text{with} \qquad a_1=1$$ the first numbers are $$\{1,1,3,14,85,621,5236,49680,521721,5994155,74701055,1003125282\}$$ which correspond to sequence $A088716$ at $OEIS$.
On this page, you will find the asymptotics $$a(n) \sim c \, n! \, n^2, \qquad \text{where} \qquad c=0.21795078944715106549282282244231982088... $$ proposed by Vaclav Kotesovec in 2014 which is incredibly closed to your approximation (look at $A238223$ for the decimal expansion of $c$).
A closed form does not seem to exist.
For $K>1$, I have not been able to find anything.