Finding value of f(n) using given binomial series

Question

I thought of multiplying the expression for (1+k)^n by itself twice and trying with coefficients but none of them can construct this series. Any hints on how to proceed?

Answer 1

Hint: Write out $(1+x)^6$ using the binomial theorem and differentiate it. What do you get?

Answer 2

So we take the expression as given and differentiate it twice and finally multiply with the initial series to get the reqd. answer.