I'm having trouble plotting the result of FindDiferrenceRoot[], for following where a->(2/3) and n from 0 to 20:
FindDifferenceRoot[Sum[(2^k*(2*k)!*HermiteH[-(2*k)-1,
a/Sqrt[2]]*((-1)^k*n!)/(k!^2*(n - k)!))/(2*k + 1), {k, 0, n}]]
Please advise.

FindDifferenceRootisn't a Mathematica function (theDifferenceRootis actually being produced bySum). Given that both of your questions so far demonstrate incorrect function usage and have only admitted answers that guess at what you really meant to do, I'd ask you to please consider what it is you're asking (and whether the answers will be useful to others in future) and, by a careful reading of the documentation, try to identify why problems might have arisen before posting further questions. Thanks for your consideration! – Oleksandr R. Aug 3 '12 at 12:29FindDifferenceRoot, it seems exceedingly unlikely that anyone else could be benefited by keeping this around for reference.) – Oleksandr R. Aug 3 '12 at 12:36FindDifferenceRoot? – acl Aug 3 '12 at 14:58