= Taylor expansion of a translated Gaussian
{title2=$e^{-(x-a)^2}=e^{-a^2}[1+2ax+(2a^2-1)x^2+(4a^3/3-2a)x^3+\cdots]$}
Multiply the two <exponential series> in $e^{-(x-a)^2}=e^{-a^2}e^{2ax}e^{-x^2}$ to obtain the displayed <Taylor expansion>. Its first three nonzero terms depend on the real parameter $a$. For $a\ne0,\pm1/\sqrt2$, they end at degree two. At $a=0$, odd coefficients vanish and the first three terms are $1-x^2+x^4/2$. At $a=\pm1/\sqrt2$, the quadratic coefficient vanishes, leaving $e^{-1/2}(1+2ax-4ax^3/3)$ as the first three nonzero terms. Parameter-dependent vanishing coefficients must be checked when a question counts nonzero terms rather than prescribing an expansion order.
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