300 Followers Problem - Combinatorial Expressions!

\[\large{R_n = \sum_{k=0}^{n} \binom{2k}{k} \quad , \quad S_n = \sum_{k=0}^{n} (-1)^k \binom{2k}{k} }\]

For \(n\) belonging to the set of non-negative integers, define \(R_n\) and \(S_n\) as of above. Also define \(A_n\) and \(B_n\) as:

  • \(\large{A_n = \displaystyle R_n^2 + 2 \sum_{k=1}^n \binom{2n+2k}{n+k}R_{n-k} }\)

  • \(\large{B_n = \displaystyle S_n^2 + 2 \sum_{k=1}^n (-1)^{n+k} \binom{2n+2k}{n+k}S_{n-k} }\)

Evaluate the value of: \(\large{\log_2\left(5B_{2015} - 3A_{2015} \right) }\) upto three correct places of decimals.

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