For any natural number $N$:
$$_t k_{mn}^1 = \frac{1}{m!n!}\frac{\partial^{n+m} k_1 (0, 0)}{\partial x^m \partial t^n}, \qquad (m,n=0,1,\ldots, N)$$
where $k_1(x,t)$ is a known function, for example $k_1(x,t) = x^5 + t^3$.
How I can write a code in Mathematica to calculate these derivatives?
I suspect that you want to evaluate the derivatives at $(0,0)$ after performing the derivative. Otherwise the whole expression would vanish anyways.
In the documnetnation for D you will find that there is indeed a way to achieve what you want: D[f[x1,...,xn],{x1,a1},...,{xn,an}] successively computed the $a_i$th derivative of f with respect to its variables.
For your case that means you can use
k1[x_, t_] := x^5 + t^3;
k[n_, m_] := (1/(m!*n!)*D[k1[x, t], {t, n}, {x, m}]) /. {x -> 0, t -> 0};
which will give you zero for this particular choice of k1.
