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in a simple sequence, example the fib sequence we have


x²-x-1 = 0

evalueting the generating function with a number, example 10

f(10) = 10² - 10 - 1 = 89

with the power sum of the fib multiplied by 10 one get the reciprocal of 89

other sequences



Would be possible do some umbral calculus with tetration indexes and series with recurrence inside recurrence ?
Check out my page on Combinatorics. My entire approach to tetration uses Umbral calculus. The following is a calculation I'm debugging now.
[Image: umbral.png]
Huh  it seems a complete area of research 

although my ignorance is pretty big in combinatorics also

Sándor, J., & Crstici, B. (2004). Stirling, bell, bernoulli, euler and eulerian numbers. Handbook of Number Theory II, 459–618. doi:10.1007/1-4020-2547-5_5
I recommend checking out Analytic Combinatorics and John Baez's writings

Umbral calculus recognizes the reality that we live in a polynomial-centric world in math, a concept aligned with n-dimensional hypercubes. But there are relationships that span the hypercube and other basic combinatorial structures. Rota reduced this to linear operator, so the weird looking efficacy of umbral calculus is legit. 

Our common interest is iterated functions. Well in combinatorial terms we want to look into the realm of compositions instead of polynomials. One important property is that recursive compositions terminate at https://oeis.org/A000311 instead of continuing to grow. The combinatorial structure A000311 is "Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n". It is also the number of ways to classify a group of objects.