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I'm optimizing for a probability distribution on non-negative integers and numerical calculations show that the distribution is supported on integers of the form $\binom{i}{2}$. In trying to figure out what the probability generating function should look like, I'm not aware of any power series that only includes quadratic powers (or, for that matter, just an infinite subset of integers that do not form an arithmetic progression). So my main question is:

Is anyone aware of any literature on power series of the form

$f(x)=\sum_{i=0}^\infty a_i x^{\binom{i}{2}}$?

Are there any known series or identities that take the above form that may guide me in the right direction?

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