MathDB

f(0)=1,f(1)=0, f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)

Source: ISI(BS) 2006 #10

June 8, 2012

functionratioprobabilityinductiongeometric sequencealgebra proposedalgebra

Problem Statement

Consider a function

f

on nonnegative integers such that

f(0)=1, f(1)=0

and

f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)

for

n \ge 2

. Show that

\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}