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Determining the Number of Solutions Using Generating Functions

  1. Mar 25, 2008 #1
    1. The problem statement, all variables and given/known data
    Determine the number of solutions in nonegative intergers to the equation:
    a + 2b + 4c = 10[tex]^{30}[/tex]

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

    3. 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?
  2. jcsd
  3. Mar 26, 2008 #2
    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)[tex]\Sigma[/tex](n+1)x^n][(1\2)[tex]\Sigma[/tex]x^n], so the coefficient for 10^30 is -1\4[((10^30)/2) +1],

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

    (9\30)[tex]\Sigma[/tex]x^n, so the coefficient is 9\32

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

    (5\32)[tex]\Sigma[/tex](-x)^n, so the coefficient is 5\32

    (1\8)[tex]\Sigma[/tex]x^(2n+1), so the coefficent 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!!!
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