Determining the Number of Solutions Using Generating Functions

[alec_tronn]

Homework Statement

Determine the number of solutions in nonegative intergers to the equation:
a + 2b + 4c = 10^{30}

Homework Equations

The generating function I've found is f(x) = 1/[(1-x^{4})(1-x^{2})(1-x)]

The Attempt at a Solution

I'm pretty sure I need to get from here to an explicit formula, but I'm not sure how to start. Any hints to get me started on this one?

 

[alec_tronn]

Alright, using the Apart function in Mathematica, I separated the generating function into:

[-1/(8(-1+x)^3)] + [1/(4(-1+x)^2)]- [9/(32(-1+x))]+ [1/(16(1+x)^2)]+ [5/(32(1+x))]+ [(1+x)/(8(1+x^2))]

Then, I turned those each into the followings infinite series:
[(-1/2)\Sigma(n+1)x^n][(1\2)\Sigmax^n], so the coefficient for 10^30 is -1\4[((10^30)/2) +1],

(-1\2)\Sigma(n+1)x^n, so the coefficient is -1\2(10^30+1)

(9\30)\Sigmax^n, so the coefficient is 9\32

(1\4)\Sigma(n+1)(-x)^n, so the coefficient is (1\4)(10^30+1)

(5\32)\Sigma(-x)^n, so the coefficient is 5\32

(1\8)\Sigmax^(2n+1), so the coefficient is 1\8,

I add up all the coefficients is (-1\2)((10^30)+1)+(1\2)

Thats negative, so it can't be the answer. I'm awfully rusty in the Calc 2 skills of making functions like that into series... so if anybody could catch any of my mistakes I'd be forever grateful!