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Let-n-be-a-positive-integer-Prove-that-k-0-n-2-k-n-k-n-k-n-k-2-2n-1-n-




Question Number 120188 by benjo_mathlover last updated on 30/Oct/20
Let n be a positive integer .  Prove that Σ_(k=0) ^n 2^k   ((n),(k) )  (((  n−k)),((⌊((n−k)/2)⌋)) ) =  (((2n+1)),((     n)) )
$${Let}\:{n}\:{be}\:{a}\:{positive}\:{integer}\:. \\ $$$${Prove}\:{that}\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\mathrm{2}^{{k}} \:\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}\:\begin{pmatrix}{\:\:{n}−{k}}\\{\lfloor\frac{{n}−{k}}{\mathrm{2}}\rfloor}\end{pmatrix}\:=\:\begin{pmatrix}{\mathrm{2}{n}+\mathrm{1}}\\{\:\:\:\:\:{n}}\end{pmatrix} \\ $$

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