upper bounding alternating binomial sums

What you ask for is the sign-alternating analogue of the Chu-Vandermonde convolution.

While I haven't proved that there is no closed formula, there is an interesting statement in the 1992 article by Andersen and Larsen, Combinatorial Summation Identities. The authors say that "the sign-alternating analogue [...] of the Chu-Vandermonde convolution is hardly known". For 3 special cases, they give the formulae by Kummer.

In his 2006 book "Summa Summarum", Larsen doesn't state new results either, so the 1992 quote may still hold.

Preserving color of text piped through "less" or "more"

Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$

identity operator isn't bounded

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Continued Fraction: Please prove $\frac{1}{e \gamma (x+1,1)}=x+\frac{1}{x+1+\frac{2}{x+2+\frac{3}{x+3+\frac{4}{\dots}}}}$

Formula for $\sum_{k=0}^n k^d {n \choose 2k}$

Show that $f$ is a polynomial if it's the uniform limit of polynomais

Making the water gallon brainteaser rigorous

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Showing that the image of a function is $\mathbb{C}$ if it satisfies a nice functional equation

Consistency of a family of probabilities in the construction of a pre-brownian motion