plz help with my maths problem

Plz help me 1.Find the closed form of the generating function A(x) for the following a_n.
(a)a_n=2^n+3^n-1
(b)a_n=1+2+⋯+n
(c)a_n=(-2)^n ((n+4)¦n)

2.Find the expansion of each of the following generating functions:
(a)(x-1)/〖(1+2x)〗^2
(b)x/(1-x^2 )
(c)(1-7x)/(1-5x+6x^2 )

3.(a)(Balanced parentheses)Let p_n be the number of ways to arrange n pairs of parentheses such that they are correctly matched.Show that {█(p_n=∑_(k=0)^(n-1)▒p_k p_(n-k-1)@p_0=1)┤.
(b)(Mountain ranges)Let m_n be the number ways to form "mountain ranges" with n upstrokes and n downstrokes that all stay above the original line.For example, when n=3,
／＼／＼／＼,

／＼
／ ＼／ ＼,

／＼
／ ＼ ／ ＼,

／＼／ ＼
／ ＼

／ ＼
／ ＼
／ ＼
Show that m_n=p_n
(c)(Diagomal-avoiding paths)Let d_n be the numner of paths on from bottom-left corner to top-right corner are there if only upward and right movements are allowed and the paths cannot go beyond the diagonal.Show that d_n=m_n.

4.Let a_n be the number of solutions of x+2y=n where x and y are non-negative integers.By using the generating function A(x) for a_n,find the general formula for a_n.