# Recursive Sum not evaluating correctly

I'm trying to evaluate the following sums that nest into eachother:

$$m_k=\frac{k}{k-1} \left(e^{\gamma}+ \sum_{i=1}^{k-2} {k-1\choose i} \frac{m_i}{i} \right) \\ m_1=e^\gamma$$ and $$\kappa_n = m_n - \sum_{i=1}^{n-1} {n-1 \choose i}m_i\kappa_{n-i} \\ \kappa_1 = m_1$$

With the following code:

moment[1] = E^(EulerGamma);
moment[k_] :=
k/(k - 1) (E^(EulerGamma) +
Sum[Binomial[(k - 1), i]*moment[i]/i, {i, 1, k - 2}])


and

cum[1] = moment[1];
cum[n_] :=
moment[n] -
Sum[moment[i]*cum[n - i]*Binomial[n - 1, i], {i, 1, n - 1}]


But it's calculating wrong on $$\kappa_4$$. Am I missing something?

The results I'm trying to replicate come from a paper that offer the first 4 values of $$\kappa$$ to be:

$$1.781072 \quad 0.3899259 \quad 0.2814155 \quad 0.2399205$$ And my results are: $$1.78107 \quad 0.389926 \quad 0.281416 \quad -3.32222$$

• Please edit your question and explain what exactly is wrong: What answer do you expect and why? Also, there is some chaos with letters, $k_{n-i}$ should perhaps be $\kappa_{n-i}$ and $k_4$ should perhaps be $\kappa_4$... Nov 6, 2022 at 21:51
• Just made the edits. Sorry for the lack of clarity. Nov 7, 2022 at 5:52
• Have you considered that the paper could be wrong? Can you provide a link (ideally open access)? Nov 7, 2022 at 5:54
• Maybe it works out fine but is moment[2] equal to $2 e^{\gamma }$ ? I ask because the Sum goes from $i=1$ to $i=k-2$ which is $i=1$ to $i=0$.
– JimB
Nov 7, 2022 at 6:12
• How to choose proper value of "n"??? Nov 7, 2022 at 7:11