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This question is motivated by Richard Stanley's A question on the Laurent phenomenon (motivated by his answer to the question what is the probability that a scissor became the champion?).

He asked about possibility to apply the same technique to other Laurent phenomenon recurrences. This question has no accepted answers, so we can try to look at this problem from another side.

Are there any more combinatorial sequences $P_n$ satisfying the following two properties:

  1. $P_n$ is a polynomial of some initial data;
  2. recurrent relation has the form $$A(P_{n+k-1},\ldots,P_n)\cdot P_{n+k}=B(P_{n+k-1},\ldots,P_n),$$ where $A\ne \rm{const}$ and $B$ are some polinomials (so $P_n$ a priori is a rational functions (of initial data) with complicated denominators).

Probably more simple examples than `scissor-the-champion' shall spread some light on this problem.

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