Let $R$ be a field, and let $(a_n)_{n\geq 0}$ be a sequence in $R$ with ordinary [generating function](/page/Generating%20Function) $A(X)=\sum_{n=0}^{\infty}a_nX^n\in R[[X]]$. For a formal [power series](/page/Power%20Series) $B(X)=\sum_{n=0}^{\infty}b_nX^n\in R[[X]]$, write $[X^n]B(X)=b_n$ for the coefficient of $X^n$ in $B(X)$. Assume that $A(X)$ is rational, so that there exist polynomials $P(X),Q(X)\in R[X]$ with $Q(0)\neq 0$ and $Q(X)A(X)=P(X)$ in $R[[X]]$.

If $A(X)\in R[X]$, then the sequence $(a_n)_{n\geq 0}$ is eventually zero. If $A(X)\notin R[X]$, then for every representation $Q(X)A(X)=P(X)$ with $P(X),Q(X)\in R[X]$ and $Q(0)\neq 0$, writing $d=\deg Q$, one has $d\geq 1$ and the sequence $(a_n)_{n\geq 0}$ eventually satisfies a linear recurrence with constant coefficients of order at most $d$: there exist an integer $N\geq 0$ and elements $r_1,\ldots,r_d\in R$ such that $a_{n+d}=r_1a_{n+d-1}+r_2a_{n+d-2}+\cdots+r_da_n$ for every integer $n\geq N$.