In this post, we prove the closed form of a nonlinear recurrence corresponding to the count of binary trees with nodes.
Posts Tagged ‘ algebra ’
Suppose that we are given the alphabet ; a word of length : , is an ordered -tuple whose elements all came from . For example, a word of length in might be the tuple , which we will hereon denote as . For some natural number , how many words of length are there that contains exactly zeros?
Suppose that we’re given the function , find the ordinary generating function associated with it in the form of . Furthermore, find/compute .
Let , where k is an integer. Prove that for any positive integer n the number
is divisible by .
Define the sequence recursively by and
Find an explicit formula for in terms of n.
Let be defined by the recurrence relation , with . Show that the expression depends only on b and , but not on a.
Find the general term of the sequence given by , and
Let . Show that for any positive integers there exists unique numbers such that the polynomial is divisible by p(x)
We derive Binet’s equation for the nth Fibonacci number as