A Binary Tree Representation
The Stirling polynomials σn(x) are related to the Bernoulli numbers by Bn = n!σn(1). S. C. Woon (Woon 1997) described an algorithm to compute σn(1) as a binary tree.
Woon's tree for σn(1) |
Woon's recursive algorithm (for n ≥ 1) starts by assigning to the root node N = . Given a node N = of the tree, the left child of the node is L(N) = and the right child R(N) = . A node N = is written as ± in the initial part of the tree represented above with ± denoting the sign of a1.
Given a node N the factorial of N is defined as
Restricted to the nodes N of a fixed tree-level n the sum of 1/N! is σn(1), thus
For example B1 = 1!(1/2!), B2 = 2!(−1/3! + 1/(2!2!)), B3 = 3!(1/4! − 1/(2!3!) − 1/(3!2!) + 1/(2!2!2!)).
Read more about this topic: Bernoulli Number
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“Everyone who enjoys supposes that the tree was concerned with the fruit, but it was really concerned with the seed.In this lies the difference between all those who create and those who enjoy.”
—Friedrich Nietzsche (18441900)