Consider the “ special function “
$$ f(z) = 1 + \sum_{n=1}^{\infty} n \frac {z^{t(n)}}{t(n)!} $$
Where $t(n)$ means n th triangular number.
I assume it has No closed form despite resembling exp and theta functions.
Although I have no proofs or reliable computations it seems this function has Some intresting properties.
It seems to be almost periodic for $re(z) > 1$. And that almost period is imaginary Just like exp !!
Also this function seems to grow like $exp( c z).$
Im fascinated. It reminds me of series multisection and substitution ( for computing integrals ).
Both on and off the real line this function behaves familiar and intresting.
But How about Some plots ( real or complex ) , proofs and ways of computing with them ?
For instance for real positive x :
$$Lim_x : f(x) \cdot exp(a x) = b $$
Does that hold and for What values $a,b$ ? ( Clearly a < 0 )
What is the value of the pseudoperiod ?
Does f(z) resemble exp rotated 3 times ?
Using $t_n=\frac{n (n+1)}{2}$
$$f(x) = 1 + \sum_{n=1}^{\infty} n \frac {x^{t_n}}{t_n!}=1+x+\frac{x^3}{3}+\frac{x^6}{240}+\frac{x^{10}}{907200}+O\left(x^{15}\right)$$ Considering $f(x)\,e^{ax}$ and developing as a Taylor series at $x=0$,we have $$f(x)\,e^{ax}=1+(a+1) x+\frac 12\left({a^2}+2a\right) x^2+\frac{1}{6} \left(a^3+3 a^2+2\right) x^3+O\left(x^4\right)$$ and then the result already given in a comment by Antonio Vargas $(a=-1,b=1$).