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\bf{PHY 742 -- Notes on Clebsch-Gordan coefficients}
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Reference: Abramowitz \& Stegum -- pg. 1006

Original formula:
\begin{eqnarray}
&<j_1 j_2 m_1 m_2|j_1 j_2 j m>=  \delta_{m,m_1+m_2} \sqrt{\frac{(j_1+j_2-j)!(j_1-j_2+j)!
(-j_1+j_2+j)!(2j+1)}{(j_1+j_2+j+1)!}}\\ \nonumber
&\times \sum_k \frac { (-1)^k \sqrt{(j_1+m_1)!(j_1-m_1)!(j_2+m_2)!(j_2-m_2)!(j+m)!(j-m)!}}{k!(j_1+j_2-j-k)!(j_1-m_1-k)!(j_2+m_2-k)!(j-j_2+m_1+k)!(j-j_1-m_2+k)!}
\end{eqnarray}

Working formula:
\begin{eqnarray}
&<j_1 j_2 m_1 m_2|j_1 j_2 j m>=  \delta_{m,m_1+m_2} \sqrt{\frac{(j_1-j_2+j)!
(-j_1+j_2+j)!(2j+1)}{(j_1+j_2-j)!(j_1+j_2+j+1)!}}\\ \nonumber
&\times \sqrt{\frac{(j_1+m1)!(j_2-m_2)!(j+m)!}{(j_1-m1)!(j_2+m_2)!(j-m)!}}
\\ \nonumber
&\times \sum_k \frac {(-1)^k }{k!} \frac{(j_1+j_2-j)!}{(j_1+j_2-j-k)!}\frac{(j_1-m_1)!}{(j_1-m_1-k)!}\frac{(j_2+m_2)!}{(j_2+m_2-k)!}\frac{(j-m)!}{(j-j_2+m_1+k)!(j-j_1-m_2+k)!} 
\end{eqnarray}
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