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Find an (approximate) closed-form solution for $S(m, b)$.

$$S(m,b)=\sum_{i=0}^{\lfloor (e-1)/2\rfloor}{e \choose i}S(m-1, b-i) \quad + \sum_{i=\lfloor (e-1)/2\rfloor+1}^{\min(b,e)}{e\choose i}{(m-1)e\choose b-i}$$

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Needs some motivation or reason anyone should care about it. Perhaps some interpretation involving a random walk or ??. –  Gerald Edgar May 15 '13 at 15:51
    
Is this maybe some type of Eulerian numbers? –  Per Alexandersson May 15 '13 at 16:00
    
@Gerald, there might be a connection with the OP's other question, mathoverflow.net/questions/130115/probability-calculation -- but if so, it would be nice to make it more explicit. –  Barry Cipra May 15 '13 at 23:24
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