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  "body": "\n## Summary\n\nThis method turns a recurrence for a sequence $(u_n)$ into an equation for its [[generating function|0PHAC833JF]] $U(x) = \\sum_{n \\ge 0} u_n x^n$.\nFor a linear recurrence with constant coefficients, the equation is linear and its solution $U$ is a rational function.\nExpanding $U$ again, for example by partial fractions, gives a closed formula for $u_n$.\nThe method is a standard tool of enumerative combinatorics and of the analysis of algorithms.\n\n## Explanations\n\n### L1 human\n\nA sequence is an infinite list of numbers, which is hard to handle at once.\nPacking the whole list into one power series turns the step-by-step rule into an algebraic equation.\nFor example, if $u_0 = 1$ and $u_{n+1} = 2 u_n$, then $U(x) - 1 = 2 x U(x)$.\nSo $U(x) = 1/(1 - 2x) = \\sum 2^n x^n$, and $u_n = 2^n$.\n",
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