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binomial_coefficient_over_the_complex_numbers [2015/12/16 10:28] nikolaj |
binomial_coefficient_over_the_complex_numbers [2016/05/12 19:27] nikolaj |
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${n\choose{k}} = \dfrac{1}{k!}\dfrac{n!}{(n-k)!}$ | ${n\choose{k}} = \dfrac{1}{k!}\dfrac{n!}{(n-k)!}$ | ||
+ | |||
+ | === Theorems === | ||
+ | ${m\choose{m}} = 1$ | ||
+ | |||
+ | ${m\choose{0}} = 1$ | ||
+ | |||
+ | == Pascal's identity (recursive formula) == | ||
+ | ${n\choose{k}} = {n\choose{k-1}} + {n-1\choose{k-1}}$ | ||
+ | |||
+ | From this it's also clear that ${n\choose{k}}$ is a sum of 1's, i.e. an integer. | ||
+ | |||
+ | <code> | ||
+ | (*Random generalization of linear such recursive schemes*) | ||
+ | |||
+ | f[m_, 0] = a[m]; | ||
+ | f[0, m_] = b[m]; | ||
+ | f[m_, m_] = c[m]; | ||
+ | |||
+ | f[n_, k_] := A*f[n, k - 1] + B*f[n - 1, k] + C*f[n - 1, k - 1] | ||
+ | |||
+ | Table[{n, k, f[n, k] // sexy}, {n, 1, 4}, {k, 1, 4}] // TableForm | ||
+ | </code> | ||
=== Reference === | === Reference === |