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Universality in Mathematica
As an example of how different primitive operations can be used to do the same computation, the following are a few ways that the factorial function can be defined in Mathematica:
f[n_] := n!

Standard examples of recursive sequences that do not come from linear recurrence relations include factorial
f[1] = 1; f[n_] := n f[n - 1]
and Ackermann functions (see below ).

functions together with factorials and multinomial coefficients then it appears that there is not.