TITLE: Generating function for reciprocals of Harmonic numbers?
QUESTION [5 upvotes]: Find an exponential or ordinary generating function of reciprocal Harmonic numbers.
$f(x)=\sum\limits_{n=1}^{\infty} \frac{1}{H_n}\frac{x^n}{n!}$ or
$f(x)=\sum\limits_{n=1}^{\infty} \frac{1}{H_n}x^n$  
Also, it would be nice to see EGF or OGF for other reciprocals of common "numbers", like binomial coefficients, Stirling numbers, Catalan numbers, etc.
P.S. I suspect one could use Digamma function here

REPLY [6 votes]: This is not a complete answer, just too long for a comment. 
Let us work with EGF $f(x)=\sum_{n=1}^{\infty} \frac{1}{H_n}\frac{x^n}{n!}$.
Clearly, $$H_n = \int_0^1 \frac{1-x^n}{1-x}dx.$$
Therefore,
$$
e^z -1 = \sum_{n=1}^\infty \frac{H_n z^n }{H_n n!} = \int_0^1 \sum_{n=1}^\infty \frac{ z^n (1-x^{n})}{H_n n!}\frac{dx}{1-x} = \\
= \int_0^1 \frac{f(z) - f(zx)}{1-x} dx = z\int_0^1 \int_{x}^1\frac{f'(zu)}{1-x}du\, dx = -z\int_0^1 f'(zu) \log(1-u) du =\\
= -\int_0^z f'(x) \log(1-x/z) dx.
$$
This gives some integral equation for $f'$.