Nonuniqueness of lognormal moments [As often happens when I have a question, in properly preparing to ask this one, I answered it myself. Nonetheless, I think this question is worth having an answer to on site]
I've read several times that "the lognormal distribution is not uniquely determined by its moments".
(For example, see the Wikipedia article on the lognormal distribution.)
This seems to be implying that there are (at least sometimes) two lognormal distributions with different parameters but the same moment-sequence.
Is that the intent? Or is it trying to suggest that there's some other distribution which has the same moment sequence?
 A: Note that the moments of the lognormal are $ {E} [X^{n}]=e^{n\mu +{\frac {n^{2}\sigma ^{2}}{2}}}$, and if you know you're dealing with a lognormal even specifying the first two moments ($n=1,2$) will identify a particular lognormal.
The actual situation intended by "the lognormal distribution is not uniquely determined by its moments" is that for every lognormal distribution, there's another not-lognormal distribution (in fact an infinity of them) with the same set of moments.
Specifically, there's a family of distributions indexed by two parameters $0<\epsilon<1$ and integer $k>0$ (that includes the lognormal as a limiting case, with $\epsilon=0$ for any $k$) which has a moment sequence that doesn't alter with $\epsilon$ or $k$. 
Stated most simply, if $f$ is a standard lognormal density ($\mu=0$, $\sigma=1$) and $g(x)=f(x)\cdot (1+\epsilon\sin[2\pi k\log(x)])$ then the contribution of $g-f$ to the $n$th moment is $0$ for each $n=1,2,...$.
(By change of variable the result extends to the whole lognormal family)
In fact Chris Heyde demonstrates this[1] for the (slightly) more general case of a shifted lognormal (three parameter lognormal; Heyde refers to it as a "general lognormal"). Again, this further extension of both $f$ and $g$ to cover the three parameter lognormal case simply involves another change of variable.
[1] Heyde, CC. (1963),
"On a property of the lognormal distribution",
Journal of the Royal Statistical Society, Series B, 25 (2): 392–393
(pdf here: http://link.springer.com/chapter/10.1007%2F978-1-4419-5823-5_6 )
