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Coefficient recursion

I was working on defining the coefficients of any \(x^{k}\), for \(k\leq n\) from the expansion of \((x+1)(x+2)\cdots (x+n)\). I created a table of the coefficients for the first few \(n\) values, where from I intended to define recursion, then a general formula. As it can be seen, there is \(n\) in the recursion, so I can't define a general member with characteristic equation.

My question is: How could I step forward? How do I define general member? And is any mistake in what I done?

\[\large K_m^{m-1} = K_{m-1}^{m-1} + m \cdot K_{m-1}^{m-2} \]

Note by Imi Mali
8 months ago

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