EViews.com

I'm not sure where to post this, but since it's a half baked suggestion I put it here. I noticed a few days ago that Normality tests included in Eviews require the series to be centered around the mode, otherwise you get rejection of H0 even when you shouldn't. So I was wondering if you could add a good-faith effort on Eviews part to try and clip the series such that only the significant part of the series is kept, in pseudo-code:
find mode
restrict sample to the x such that x >=mode/1000
perform Normality test
reset sample to what it was before

I believe Stata does its tests requiring the cumulated series, not sure if this helps solve the problem of asymmetric tails.
Example of strange test results in Eviews 8. I have a Binomial, 80% of a 0 result and 20% of a strictly positive result, repeat 30 times, I get a series that I call N30. I call N30trunc the hand-truncated version around the mode, and Norm01 is a short normal slightly off-center. The regression shows that the binomial is essentially Normal. Thank you for all your efforts, I really enjoy using Eviews.

To be honest, I'm not sure that I understand the question, nor of purpose of the proposed test.

Symmetry is a property of the normal distribution and in general I would want data that exhibited severe asymmetry to fail normality tests. Beyond that, just by looking at kernel plots of the data for each of your two series, it is not surprising that they fail normality tests.

More directly to the point, how were your data data generated and why would you expect them to be normally distributed? I am not certain why you believe that the regression results offer evidence of normality.

Thanks for the nice words about EViews...

Hmm I haven't studied the nuts and bolts of any normality test, but from the name I'd have thought they would test the gaussian shape of whatever is fed into the test. I certainly didn't think of it as a symmetry test.
I generated the series by, for each value i from 1 to 30, computing the combinations (30,30-i)*k*i*pow(0.8,30-i)*pow(0.2,i), k is a constant set at 0.2 in my case, so a binomial multiplied by k*i.
The regression shows an R2 of close to one on my data points regressed on a gaussian. The sample is moderately big. Big enough for people to say that Student t distributions at that size look like normals. The function is well-behaved, not something that would make me worry like say 1/x*sin(1/x) in the vicinity of zero (I'm sure one could extract specific points from a function of that family, or perhaps a slightly different function, to make a plot that could be taken for a gaussian; but I'm only interested in standard stuff so far), so from all that precedes I conclude the shape is close to Normal. A simple plot shows that's a reasonable conclusion.

On the other hand if the test is doing something based on symmetry, it is fairly obvious it would fail. But I wouldn't call it a Normality test then, although at a quick googling, I now know it's not just Eviews doing that.
Perhaps I should develop my own test, something based on the Taylor expansion of the exponential and that doesn't rely on symmetry. Regardless, thank you for your time.

I think you've confused generating numbers from a normal distribution with generating the height of a normal pdf. It looks like you have the latter, which are not even vaguely normal. To see this, look at the histogram for your series.