Sorry. The Catalan numbers are 1,1,2,5,14,42,... given by the formula Cn=(1/(n+1))*2n choose n. With 2n choose n equal to (2n)!/(n!n!). Sorry for the bad form, I don't have any math writing program.
Homework Statement
I need to prove two things about the Catalan numbers. The first is that Cn is odd iff n=(2^k)-1 for some positive integer k.
The second is that given the matrix A defined by the rule a(i,j)=C(i+j), prove that det A=1. I have not covered determinants in my linear class...
Homework Statement
I need to find some huge numbers and don't know how to do it using a computer. Everything that I have tried doesn't work because the numbers are too big.
Homework Equations
The equations are 1/((1-x)(1-x^2)(1-x^3)) and 1/((1-x)(1-x^3)(1-x^4)) where x=10^30.
The...
Homework Statement
I am trying to find the number of nonnegative integer solutions to a+2b+4c=10^30. I found a generating function, and need to check the coefficient of 10^30.
Homework Equations
The generating function is 1/((1-x)(1-x^2)(1-x^4)). I found the PFD, which is...
Homework Statement
I need to find the number of nonnegative integer solutions to the equation a+2b+4c=10^30
Homework Equations
The Attempt at a Solution
I was thinking of trying to find perhaps some sort of a multinomial generating function, but am not sure how that will help...
Homework Statement
I need to find the number of ways to make change for a dollar using an even number of coins, using only pennies, nickels, dimes, quarters, and fifty cent pieces.
Homework Equations
I found the generating function for the ways of making change for a dollar...
Homework Statement
The question is to evaluate the infinite series of the Sum[(((-1)^n)*a(n))/10^n], as n goes from zero to infinity, and a(n) is the recurrence relation a(n)=5a(n-1)-6a(n-2) where a(0)=0, and a(1)=1
Homework Equations
I found the explicit equation for a(n)=3^n - 2^n...