A binomial sum

[Henry Gould]

Well the original question actually comes down to this:

Can we find a closed formula for the binomial coefficient summation
sum(m=0,n,(-1)^m*binomial(n+m,m))?
The answer is NO.

If you omit the alternating signs (-1)^m then it can be done at once of
course . . .
See formula No. (1.49), page 7 in my book "Combinatorial Identities, A
Standardized Set of Tables Listing 500 Binomial Coefficient Summations"
which I published here in Morgantown, West Virginia in 1972. (By the way my
book is still available in print)

Henry Gould