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Prove-k-0-m-m-k-2-n-2m-k-2m-m-n-n-2-




Question Number 154928 by qaz last updated on 23/Sep/21
Prove::   Σ_(k=0) ^m  ((m),(k) )^2  (((n+2m−k)),((      2m)) )= (((m+n)),((    n)) )^2
$$\mathrm{Prove}::\:\:\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{m}} {\sum}}\begin{pmatrix}{\mathrm{m}}\\{\mathrm{k}}\end{pmatrix}^{\mathrm{2}} \begin{pmatrix}{\mathrm{n}+\mathrm{2m}−\mathrm{k}}\\{\:\:\:\:\:\:\mathrm{2m}}\end{pmatrix}=\begin{pmatrix}{\mathrm{m}+\mathrm{n}}\\{\:\:\:\:\mathrm{n}}\end{pmatrix}^{\mathrm{2}} \\ $$

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