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Mathematica evaluates Sum[((n - y - 1)*(n - y)^2*n^y)/y!, {y, 0, n - 2}] as -2 e^n n. This should not be a negative value. What am I doing wrong?

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This is a problem of version 8. In version 9.04 I get (1/(n! Gamma[ n]))(-n! (5 n^n + E^n (1 + 4 n) Gamma[n] - E^n Gamma[n, n] - 2 E^n Gamma[1 + n, n]) + n^n Gamma[n] HypergeometricPFQ[{2, 2, 2}, {1, 1, 1 + n}, n]). –  Sjoerd C. de Vries Mar 11 '13 at 22:34
    
OK. HypergeometricPFQ[{2, 2, 2}, {1, 1, 1 + n}, n] looks very hard to interpret! Do you have any idea what that looks like asymptotically? –  Lembik Mar 11 '13 at 22:38
    
@Sjoerd Interestingly, if you simply change the upper limit from n-2 to n-1 (which introduces a zero into the sum, leaving it unaltered) MMA 8 obtains (1/((1 + n)! Gamma[ 2 + n]))n ((1 + n)! (n^n (1 + 2 n) - E^n Gamma[2 + n] - E^n Gamma[2 + n, n]) + n^n Gamma[2 + n] HypergeometricPFQ[{2, 2, 2}, {1, 1, 2 + n}, n])--which numerical tables show is correct. Also, if the full Sum is involved within a more complex expression, it might get evaluated correctly. –  whuber Mar 11 '13 at 22:39
    
Lembik, hypergeometric functions are nice: they have series expansions, known poles, integral representations, etc. Take a look at the help page for the series definition. –  whuber Mar 11 '13 at 22:40
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@whuber Fact remains that this particular result is indeed dead wrong. –  Sjoerd C. de Vries Mar 11 '13 at 22:40
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1 Answer

up vote 2 down vote accepted

This is apparently a problem of version 8 where I get the same. In version 9.0.1 I get

(1/(n! Gamma[ n]))(-n! (5 n^n + E^n (1 + 4 n) Gamma[n] - E^n Gamma[n, n] - 
2 E^n Gamma[1 + n, n]) + n^n Gamma[n] HypergeometricPFQ[{2, 2, 2}, {1, 1, 1 + n}, n])

or, with nicer formatting,

Mathematica graphics

This can be reduced with

Sum[((n - y - 1)*(n - y)^2*n^y)/y!, {y, 0, n - 2}] // FullSimplify

to

(2 n (n^n - E^n Gamma[n, n]))/Gamma[n]

or, in TraditionalForm,

Mathematica graphics

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That's very clever of Mathematica! –  Lembik Mar 11 '13 at 22:54
    
WA online gives the same error. Wouldn't they be using the most recent version? –  Lembik Mar 12 '13 at 18:38
    
@Lembic I seem to recall reading that W|A is updated frequently, so indeed that is what one would expect. –  Sjoerd C. de Vries Mar 12 '13 at 21:25
    
I mean W|A gives the wrong answer where apparently 9.0.1 gives the right answer. I don't have 9.0.1 to test sadly. On a related topic, do Wolfram release bug fixes for Mathematica or do you have to wait another year to get an upgrade? –  Lembik Mar 12 '13 at 22:02
    
@lembik I understood what you meant, and I find it curious as well. As to fixes: WRI releases only a few updates for each major version. Generally I'd say two or three. Depending on your license you may or may not have access to them. –  Sjoerd C. de Vries Mar 13 '13 at 8:22
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