generating-function

Which one of the following is a closed form expression for the generating function of the sequence $$\left\{ {{a_n}} \right\},$$ where $${a_n} = 2n + 3$$ for all $$n = 0,1,2,....?$...

The coefficient of $${x^{12}}$$ in $${\left( {{x^3} + {x^4} + {x^5} + {x^6} + ...} \right)^3}\,\,\,\,\,\,$$ is _____________.

What is the generating function G (z) for the sequence of Fibonacci numbers?

(a) Solve the recurrence equations $$\,\,\,\,\,\,\,\,\,T\left( n \right) = T\left( {n - 1} \right) + n$$ $$\,\,\,\,\,\,\,\,\,T\left( 1 \right) = 1T$$ (b) What is the generating fun...