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  1. Path: sparky!uunet!usc!cs.utexas.edu!sun-barr!ames!purdue!mentor.cc.purdue.edu!pop.stat.purdue.edu!hrubin
  2. From: hrubin@pop.stat.purdue.edu (Herman Rubin)
  3. Newsgroups: sci.math
  4. Subject: Summation of "finite difference powers"
  5. Message-ID: <54641@mentor.cc.purdue.edu>
  6. Date: 21 Jul 92 14:15:48 GMT
  7. References: <1092@kepler1.rentec.com>
  8. Sender: news@mentor.cc.purdue.edu
  9. Organization: Purdue University Statistics Department
  10. Lines: 26
  11.  
  12.  
  13. >Recently, there was a question about 'molecules' which really asked for
  14. >the sum of the first N triangular numbers, where N=10^8. I misread the
  15. >problem and posted the N_th triangular number. the correct solution is:
  16.  
  17. >     N
  18. >    sum k*(k+1)/2 = S_N
  19. >     1
  20.  
  21.             .....................
  22.  
  23. The various responders seem to be ignorant of the "standard" 
  24. result, easily proved by induction, that
  25.  
  26.      N
  27.     sum k*(k+1)* ... *(k+j-1) = k*(k+1)* ... *(k+j)/(j+1)
  28.      1
  29.  
  30. This is the finite difference analog of int(x^j) = x^(j+1)/(j+1),
  31. and the extensions to the gamma functions versions are also valid,
  32. and likewise proved by induction.
  33. -- 
  34. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
  35. Phone: (317)494-6054
  36. hrubin@pop.stat.purdue.edu (Internet, bitnet)  
  37. {purdue,pur-ee}!pop.stat!hrubin(UUCP)
  38.