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m Fixed sections for the Buzen's algorithm bit |
Added an alternative (more general?) derivation of the recursive relationship |
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Line 23:
<math>g(m,n) = \sum_{n_1 + n_2 + \cdot\cdot\cdot + n_m = n} \prod_{i=1}^m f_i( n_i ) = \sum_{n_1 + \cdot\cdot\cdot + n_{m-1}=n, n_m=0} \prod_{i=1}^{m-1} f_i( n_i ) + \sum_{n_1 + \cdot\cdot\cdot + n_m=n, n_m>0} \prod_{i=1}^{m} f_i( n_i ) = g(m-1, n) + \frac{f_m( n )}{f_m( n-1 )}g(m, n-1)</math> (Is this last bit true?)
Alternative:
<math>f_m( 0 ) \sum_{n_1 + \cdot\cdot\cdot + n_{m-1}=n, n_m=0} \prod_{i=1}^{m-1} f_i( n_i ) + \sum_{n_1 + \cdot\cdot\cdot + n_m=n, n_m>0} \prod_{i=1}^{m} f_i( n_i ) = f_m( 0 )g(m-1, n) + \frac{f_m( n )}{f_m( n-1 )}g(m, n-1)</math>
<math>g(m,0) = 1</math>, <math>g(1, n) = f_1(n)</math>
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