Another Binomial Madness

Calculus Level 5

\[ (1+x)^n=C_0+C_1x + C_2x^2 + \cdots + C_rx^r + \cdots + C_n x^n \]

Let \(n\) be a positive integer such that the above shows an algebraic identity such that the variables \(C_0, C_1, \ldots, C_n\) are independent of \(x\).

\[\large \dfrac{C_0}{1} - \dfrac{C_1}{8} + \dfrac{C_2}{15} - \cdots +(-1)^n \dfrac{C_n}{7n+1} \]

If the closed form of the expression above can be expressed as \[ \dfrac{a \cdot 7^{n+b} \cdot (n+c)!}{1\cdot8\cdot15\cdots(7n+1)} \; ,\] where \(a,b\) and \(c\) are non-negative integers, find \(a+b+c\).

\(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

Click here for Part I of the problem.

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