umbral and tetration tetration101 Junior Fellow  Posts: 6 Threads: 4 Joined: May 2019 03/01/2020, 10:47 PM http://mathworld.wolfram.com/GeneratingFunction.html in a simple sequence, example the fib sequence we have https://en.wikipedia.org/wiki/Fibonacci_number 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 https://en.wikipedia.org/wiki/Hofstadter_sequence http://mathworld.wolfram.com/MallowsSequence.html https://en.wikipedia.org/wiki/Umbral_calculus https://en.wikipedia.org/wiki/1_%2B_2_%2..._summation Would be possible do some umbral calculus with tetration indexes and series with recurrence inside recurrence ? Daniel Fellow   Posts: 90 Threads: 33 Joined: Aug 2007 03/02/2020, 12:05 AM (This post was last modified: 03/02/2020, 02:01 AM by Daniel.) Check out my page on Combinatorics. My entire approach to tetration uses Umbral calculus. The following is a calculation I'm debugging now.  tetration101 Junior Fellow  Posts: 6 Threads: 4 Joined: May 2019 03/02/2020, 05:19 AM 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 Daniel Fellow   Posts: 90 Threads: 33 Joined: Aug 2007 03/02/2020, 06:28 AM 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. « Next Oldest | Next Newest »