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I have a generating function

$$\sum_{m,n} P(m,n)y^{m}x^{n} = \prod(1-yx^t)^{-1}$$

where $t \in \{1,2,3,5,8,...\}$ i.e. Fibonacci set with single 1.

For now, I have a naive script to calculate coefficient of $y^{m}x^{n}$ which works by expanding and ignoring higher order terms but this gets slow with $n$ greater than $10^{6}$

I saw various coefficient derivations for generating functions in one variable, however I am unable to get a clean form in terms of combinations for the above generating function.

Is it possible to get a function(m,n) for the coefficients?

EDIT : The given generating function calculates how many ways to add up to n using exactly m numbers from the set

chiragjn
  • 103
  • $P(m,n)$ counts the number of ways in which $n$ can be written as the sum of $m$ Fibonacci numbers. There's quite a bit of information about partitions into Fibonacci numbers in the paper Fibonacci partitions (Neville Robbins, The Fibonacci Quarterly, Vol. $34$ ($1996$), p. $306$-$313$), but I didn't find anything relating to partitions into a specified number of Fibonacci numbers. See also http://math.stackexchange.com/questions/1731714. – joriki May 08 '16 at 13:10
  • I know what the generating function calculates and got to this generating function,but unable to get any further because I am unable to derive any function for the coefficients – chiragjn May 08 '16 at 13:14
  • If you already knew that this is what $P$ counts, you could have included that information in the question. – joriki May 08 '16 at 13:14

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