scholarly journals Counting ternary trees according to the number of middle edges and factorizing into (3/2)-ary trees

2022 ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Integer Sequences ◽  
Cubic Equation

The sequence A120986 in the Encyclopedia of Integer Sequences counts ternary trees according to the number of edges (equivalently nodes) and the number of middle edges. Using a certain substitution, the underlying cubic equation can be factored. This leads to an extension of the concept of (3/2)-ary trees, introduced by Knuth in his christmas lecture from 2014.

A New Solution of the Cubic Equation

1898 ◽  
Vol 5 (2) ◽  
pp. 38 ◽  
Author(s):  
L. E. Dickson
Keyword(s):  
Cubic Equation

scholarly journals The Joukowsky Map Reveals the Cubic Equation

2019 ◽  
Vol 126 (1) ◽  
pp. 33-40
Author(s):  
Javier Sánchez-Reyes
Keyword(s):  
Cubic Equation

INDEX-DEPENDENT DIVISORS OF COEFFICIENTS OF MODULAR FORMS

2013 ◽  
Vol 09 (07) ◽  
pp. 1841-1853 ◽  
Author(s):  
B. K. MORIYA ◽  
C. J. SMYTH
Keyword(s):  
Prime Number ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Survey Methods ◽  
Integer Sequences ◽  
Positive Integers

We evaluate [Formula: see text] for a certain family of integer sequences, which include the Fourier coefficients of some modular forms. In particular, we compute [Formula: see text] for all positive integers n for Ramanujan's τ-function. As a consequence, we obtain many congruences — for instance that τ(1000m) is always divisible by 64000. We also determine, for a given prime number p, the set of n for which τ(pn-1) is divisible by n. Further, we give a description of the set {n ∈ ℕ : n divides τ(n)}. We also survey methods for computing τ(n). Finally, we find the least n for which τ(n) is prime, complementing a result of D. H. Lehmer, who found the least n for which |τ(n)| is prime.

Sign in / Sign up

Export Citation Format

Share Document