X n e 1 2 n x so for all k the egf is e x 1 k the

This preview shows page 11 - 12 out of 12 pages.

+ _+ x n! +_ = e -1 2 n x so for all k the e.g.f. is (e x - 1) k . The coefficient of n x n! in (e x - 1) k is precisely k! S(n,k); we can compute it as follows: i=0 k i (k-i)x x k k i (-1) e = (e -1) = H(x) . Substitute (k - i)x for x in the usual series for e x = n 0 n x n! to get H(x) = i=0 k i n 0 n k i (-1) 1 n! (k- i) xSUPn = n 0 n i=0 k i n x n! (-1) k i (k- i)
Image of page 11

Subscribe to view the full document.

July 30, 1996 12 k! S(n,k) = i=0 k i n (-1) k i (k- i) S(n,k) = 1 (-1 k (k- i i=0 k i n Exercise Find the number of r-digit quatenary requences (digits 0,1,2,3) with an even number of zeros. Off number of 1's. Egf for 0's egf for 1's = = = 1 4 4 = 4 if r > 0 r r-1 Simple form combinatorial argument exists. Can you find one?
Image of page 12
You've reached the end of this preview.
  • Fall '06
  • miller
  • Generating function, xk, e.g.f.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern