So in this equation:
The number of connected labeled bipartite graphs with bipartition (X,Y) where |X|=m and |Y|=n is the coefficient of $x^my^n/m!\,n!$ in $$\log\biggl(\sum_{m,n=0}^\infty 2^{mn} \frac{x^m}{m!}\frac{y^n}{n!}\biggr).$$
I am not exactly sure what is going on here.
The equation is supposed to yield this sequence:
A123260
For example T(2, 2) = 5.
So I try
for (m,n) =(1,1),(1,2),(2,1),(2,2)
$$
2^{mn} * \frac{x^m}{m!}*\frac{y^n}{n!}$$
and then add all of the products and get 26. Log(26) does not equal 5. Could someone please explain to me what I am doing wrong?
You have a power series $A(x)$ with known coefficients and with constant term 1. You want the coefficients of the series $B(x)=\log(A(x))$. Taking derivatives yields $$ B'(x) = A'(x)/A(X) $$ and hence $$ B'(x)A(x) = A'(x) $$ Assume the coefficients of $x^k$ in $A(x)$ and $B(x)$ are $a_k/k!$ and $b_k/k!$. Then the coefficient of $x^k$ in the LHS is $$ \sum_{i=0}^k \frac{b_{i+1}}{i!} \frac{a_{k-i}}{(k-i)!} $$ and in the RHS it is $$ \frac{a_{k+1}}{k!}. $$ This gives us a recursion: we know $b_0=0$ and $b_1=1$ and if we have $b_1,\ldots,b_k$, we can compute $b_{k+1}$.