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I have encountered the following sum:

$\sum_{n=0}^{\infty}\frac{t^{n}}{n!} \left(\frac{d^{2}}{dx dy}\right)^{n}e^{txy},$

where $t$, $x$ and $y$ are real numbers. The differentiation results in some family of polynomials which I could not identify, so the sum could be written like

$e^{txy} \sum_{n=0}^{\infty}\frac{t^{n}}{n!} t^n P_n(t x y).$

First few polynomials are $1$, $x+1$, $x^2+4x+2$, $x^3+9x^2+18x+6$.

Is there some way to evaluate the sum? More generally, what are the ways to do that for some known polynomial families (maybe I can do it by analogy)?

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