# Generating Function with lambda terms

• Mar 21st 2013, 08:56 AM
schreckenstat
Generating Function with lambda terms
I've been working on a homework problem for an enumeration course that has been giving me fits.

Find a formula for f(n) if $\lambda(i) = \lfloor|sin(5i)|\rfloor\ \forall i \in N$.

$\Lambda (y) = \sum_{k\geq0}\lambda(k)y^k$

$F(x) = \Lambda (F(x))$

$F(x) = \sum_{i\geq0}\lfloor|sin(5i)|\rfloor F^i(x)$

Can anyone explain how I can find a power series representation of F(x) from which I can find f(n)?

Thanks.
• Mar 22nd 2013, 04:29 AM
chiro
Re: Generating Function with lambda terms
Hey schreckenstat.

What is F^i(x) referring to? Are these derivatives of some kind?
• Mar 22nd 2013, 02:49 PM
schreckenstat
Re: Generating Function with lambda terms
Sorry, I probably should've been a bit more descriptive in my first post. $F^i(x)$ would be the function $F(x)$ to the ith power. Generally in these sort of problems you solve for the generating function F(x), and then find a power series representation for F(x) from which you can construct another formula f(n) which would solve a recurrence relation. An easier example might be something like if $\lambda(0)=1\ ,\ \lambda(1)=2\ , \lambda(2)=1$, using the first two formulas in the first post we would have $F(x)=x(1+2F(x)+F^2(x))$. In this case you could use the quadratic equation to find an expression for F(x), and then find a power series representation of F(x) to find f(n). The problem I'm having with this one is that I'm not sure how to attack that sum of sines to find an expression for F(x). Hope that makes things a bit more clear.
• Mar 22nd 2013, 04:59 PM
chiro
Re: Generating Function with lambda terms
I don't think i can help you in this case unfortunately. This is beyond my current set of knowledge at this time.