The Two who Reccur

Discrete Mathematics Level 4

Let there be two sequences defined as

{\(a_n\)} =\( \large \sum_{j=1}^n \sum_{k=j}^n \dbinom n k \dbinom k j\)

and {\(b_n\)} = {\(a_n\)} +\(2^{n+1}\)

Then {\(b_{n+1}\)} can be given by the following recurrence formula

{\(b_{n+1}\)} = \(\alpha\) {\(b_n\)} + \(\beta\) {\(b_{n-1}\)} where \(\alpha\) and \(\beta\) are constant integers. Find \(\alpha\) + \(\beta\)

Hint: {\(a_{n+1}\)} can also be given by the same recurrence formula


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