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Central Binomial Coe¢ Cients
| Content Provider | Semantic Scholar |
|---|---|
| Author | Finch, Steven |
| Copyright Year | 2007 |
| Abstract | The largest coe¢ cient of the polynomial (1 + x) n is [1] A(n) = n bn=2c = n dn=2e : It possesses recursion n + 1 2 A(n + 1) = (n + 1)A(n); A(0) = 1 and asymptotics A(n) r 2 n 1=2 2 n as n ! 1. Another interpretation of A(n) is as the number of sign choices + and such that 8 < : 1 1 1 1 = 0 if n is even, 1 1 1 1 | {z } n = 1 if n is odd. The latter is an especially attractive characterization of the n th central binomial coe¢ cient. Contrast this with the n th central trinomial coe¢ cient, B(n), de…ned to be the largest coe¢ cient of the polynomial (1 + x + x 2) n. There is no simple closed-form expression for B(n) [2]. It possesses recursion (n + 1)B(n + 1) = (2n + 1)B(n) + 3n B(n 1); B(0) = B(1) = 1 and asymptotics |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.people.fas.harvard.edu/~sfinch/csolve/cbc.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
