mmiguel Posted October 21, 2012 Report Share Posted October 21, 2012 Find a closed-form expression for F(a,b) where: F(a,b) = F(a-1,b) + F(a-1,b-1), F(a,0) = 1 for all a F(0,b) = 0 for all b a and b are positive integers Quote Link to comment Share on other sites More sharing options...

0 Anza Power Posted October 21, 2012 Report Share Posted October 21, 2012 (edited) What does F(0,0) evaluate to? you need it for F(1,1) In case F(0,0)=1 then F(a,b) = b choose a, as in the binomial coefficient between b and a. In case F(0,0)=0 then it's the binomial coefficient between b-1 and a. Given F(0,0) there is only one solution, you can see this if you draw an xy axis and mark a on the x axis and b on the y axis, then mark 0's on the a axis and 1's on the b axis, then the value of every other point on the plane is the sum of values of the point to it's left and the point under that... Edited October 21, 2012 by Anza Power Quote Link to comment Share on other sites More sharing options...

0 mmiguel Posted October 21, 2012 Author Report Share Posted October 21, 2012 What does F(0,0) evaluate to? you need it for F(1,1) In case F(0,0)=1 then F(a,b) = b choose a, as in the binomial coefficient between b and a. In case F(0,0)=0 then it's the binomial coefficient between b-1 and a. Given F(0,0) there is only one solution, you can see this if you draw an xy axis and mark a on the x axis and b on the y axis, then mark 0's on the a axis and 1's on the b axis, then the value of every other point on the plane is the sum of values of the point to it's left and the point under that... F(0,0) = 1 Sorry for leaving that out. Nice work, you are correct! Quote Link to comment Share on other sites More sharing options...

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## mmiguel

Find a closed-form expression for F(a,b) where:

F(a,b) = F(a-1,b) + F(a-1,b-1),

F(a,0) = 1 for all a

F(0,b) = 0 for all b

a and b are positive integers

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