Inspired of course by What's a natural candidate for an analytic function that interpolates the tower function? I am minded to ask what looks to me like a more natural question: given a sequence $a_1,a_2,a_3,\ldots$ of complex numbers, is there always a holomorphic function $f$ defined on the entire complex plane, with $f(n)=a_n$ for $n=1,2,3,\ldots$? No idea what the answer is myself, but wouldn't surprise me if it were wellknown and even easy.

This is Exercise 6, Page 26, of Knopp's Problem Book in the Theory of Functions, Volume 2: For any sequence of complex numbers $z_n$ with no finite limit point, and for any sequence of complex numbers $w_n$, there is an entire function mapping $z_n$ to $w_n$. The proof goes like this: Use the Weierstrass Factor Theorem to construct a function $W$ with simple zeros at the $z_n$. Use the MittagLeffler theorem to construct a function $M$ with simple poles at the $z_n$ with residues $\frac{w_n}{W'(z_n)}$. Then the function $W\cdot M$ does the job. 


Probably wellknown. Easy? I'd venture to guess that an expression like $$\sum_{n=1}^\infty b_n\frac{e^{c_n(zn)}}{(n1)!}\prod_{k=1}^{n1}(zk)$$ can be made to work. You'll have to pick the $b_n$ successively to make the $N$'th partial sum equal to $a_N$, and real constants $c_n$ large enough to obtain uniform convergence to the left of any fixed vertical line. E.g., so that the $n$'th term has absolute value less than $2^{n}$ when $\operatorname{Re} z<n/2$. 

