I'm trying to prove that:
$$\kappa_{k_1,\dots,k_n}(X_1,\dots,X_n)=\kappa_{\underbrace{1,\dots,1}_{k_1+\dots+k_n}}(\underbrace{X_1,\dots,X_1}_{k_1},\dots,\underbrace{X_n,\dots,X_n}_{k_n})$$
where $X_1,\dots,X_n$ are random variables, $k_1,\dots,k_n \in \mathbb{N}$, and $\kappa_{k_1,\dots,k_n}(X_1,\dots,X_n)$ is the joint cumulant of $X_1,\dots,X_n$ (namely the multidimensional cumulant), definded by:
$$\kappa_{k_1,\dots,k_n}(X_1,\dots,X_n) \equiv \partial_{t_1}^{k_1}\dots\partial_{t_n}^{k_n} K_{X_1,\dots,X_n}(t_1,\dots,t_n)\big|_{t_1,\dots,t_n=0}$$
where $K_{X_1,\dots,X_n}$ is the joint cumulant generating function. I searched on Wikipedia (https://en.wikipedia.org/wiki/Cumulant) founding that we can write:
$$\kappa_{k_1,\dots,k_n}(X_1,\dots,X_n) \equiv \partial_{t_1}^{k_1}\dots\partial_{t_n}^{k_n} K_{X_1,\dots,X_n}(t_1,\dots,t_n)\big|_{t_1,\dots,t_n=0}= \\ \\ =\partial_{t_{1,1}}\dots\partial_{t_{1,k_1}}\dots\partial_{t_{n,1}}\dots\partial_{t_{n,k_n}} K_{X_1,\dots,X_n}\left(\sum_{i=1}^{k_1} t_{1,i},\dots,\sum_{i=1}^{k_n} t_{n,i}\right)\Bigg|_{t_{i,j}=0}$$
Wikipedia tells that this proves the desired result, but I don't see the final step, namely:
$$\partial_{t_{1,1}}\dots\partial_{t_{1,k_1}}\dots\partial_{t_{n,1}}\dots\partial_{t_{n,k_n}} K_{X_1,\dots,X_n}\left(\sum_{i=1}^{k_1} t_{1,i},\dots,\sum_{i=1}^{k_n} t_{n,i}\right)\Bigg|_{t_{i,j}=0} \stackrel{?}{=} \\ \\ \stackrel{?}{=} \partial_{t_{1,1}}\dots\partial_{t_{1,k_1}}\dots\partial_{t_{n,1}}\dots\partial_{t_{n,k_n}} K_{X_1,\dots,X_1,\dots,X_n,\dots,X_n}(t_{1,1},\dots,t_{1,k_1},\dots,t_{n,1},\dots,t_{n,k_n})\big|_{t_{i,j}=0} \equiv \\ \\ \equiv \kappa_{\underbrace{1,\dots,1}_{k_1+\dots+k_n}}(\underbrace{X_1,\dots,X_1}_{k_1},\dots,\underbrace{X_n,\dots,X_n}_{k_n})$$
How can I prove this? Any hint or any source (or alternative proofs) are welcome. Thanks!