In their classical paper on fluctuations in coin tossing [On Fluctuations in Coin-Tossing](https://www.pnas.org/content/35/10/605), Chung and Feller give a precise formula for the conditional probability of the number of positive “sides” of a random walk with an even number of steps, given a particular outcome for the endpoint. 
$$
\mathbf{P}\left(\sum_{j=1}^{2n} \mathbf{1}_{(0,\infty)}\left(\frac{\mathsf{X}_{j-1}+\mathsf{X}_j}{2}\right)=2k\, \middle|\, \mathsf{X}_{2n}=2\ell\right)\,
=\, \frac{\ell}{\binom{2n}{n-\ell}}\, \sum_{i=\ell}^{k} \binom{2i}{i-\ell} \binom{2n-2i}{n-i}\, \cdot \frac{1}{i(n-i+1)}
$$
for a simple random walk $(\mathsf{X}_0,\mathsf{X}_1,\dotsc,\mathsf{X}_{2n})$.

For their Theorem 1 
$$
\mathbf{P}\left(\sum_{j=1}^{2n} \mathbf{1}_{(0,\infty)}\left(\frac{\mathsf{X}_{j-1}+\mathsf{X}_j}{2}\right)=2k\right)\,
=\, u_{2k} u_{2n-2k}\, ,\ \text{ for }\ u_{2k}\, =\, \mathbf{P}(\mathsf{X}_{2k}=0)\, =\, \frac{1}{2^{2k}} \binom{2k}{k}\, ,
$$
the generalization to odd times was performed somewhat recently by Gessel in slides

[Chung–Feller Theorems](https://people.brandeis.edu/~gessel/homepage/slides/chung-feller-slides.pdf) last few slides,

and by Grünbaum in an article 

[A Feynman–Kac approach to a paper of Chung and Feller on fluctuations in the coin-tossing game](https://arxiv.org/abs/1810.06092v1)

published in [Proceedings of the American Mathematical Society](https://doi.org/10.1090/proc/14758). The answer for that generalization is 
$$
\mathbf{P}\left(\sum_{j=1}^{2n+1} \mathbf{1}_{(0,\infty)}\left(\frac{\mathsf{X}_{j-1}+\mathsf{X}_j}{2}\right)=2k+1\right)\,
=\, u_{2k} u_{2n-2k} \cdot \frac{2k+1}{2(n+1)}\, .
$$
Incidentally, the problem was apparently also stated as an exercise in McKean's textbook: 

[Probability: The classical limit theorems](https://www.cambridge.org/us/academic/subjects/mathematics/abstract-analysis/probability-classical-limit-theorems?format=HB&isbn=9781107053212) Exercise 3.4.2 on p139.

But to the best of my knowledge nobody has stated a generalization of Chung and Feller's conditional distribution results in Theorem 2 to odd numbers of steps. (I might have missed it somewhere.) 

*Note the analog of their Theorem 2a was the limit/specialization of their Theorem 2 to $\ell=0$ giving a uniform distribution. It makes sense from their Theorem 2a formula if you cancel the $\ell$ in the numerator outside their sum with the $i$ (which is necessarily $0$ in the single-summand sum) in the denominator inside the sum.*

As a secondary question, I wonder why combinatorialists sometimes derive useful formulas and advertise them as Gessel (and possibly McKean?) did, but then do not publish them. I have asked that question specifically over at AcademiaSE with more references for an example related to this question:

[Is it common for combinatorialists to not publish all their results? If so, why?](https://academia.stackexchange.com/questions/180689/is-it-common-for-combinatorialists-to-not-publish-all-their-results-if-so-why)