Converting a recursive definition to an explicit one - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:22:31Z http://mathoverflow.net/feeds/question/97450 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97450/converting-a-recursive-definition-to-an-explicit-one Converting a recursive definition to an explicit one unknown (google) 2012-05-20T02:05:17Z 2012-05-20T02:20:52Z <p>Is there an explicit form for $a_x$ (whole numbers x) given that $a_x = \displaystyle\sum_{i=1}^{x-1} \binom{x-1}{i} a_i$?</p> <p>I've listed out the first few terms:</p> <p>for $x=0,1,2,3,4,5,6, 7$</p> <p>we have $a_x =1, 1, 2, 5, 15, 52, 203, 877$ respectively which shows no obvious pattern, except growing extremely quickly.</p> http://mathoverflow.net/questions/97450/converting-a-recursive-definition-to-an-explicit-one/97452#97452 Answer by Mark Sapir for Converting a recursive definition to an explicit one Mark Sapir 2012-05-20T02:12:40Z 2012-05-20T02:20:52Z <p>These are Bell numbers, <a href="http://oeis.org/search?q=1%2C1%2C2%2C5%2C15%2C52&amp;language=english&amp;go=Search" rel="nofollow">A000110</a>. The relatively explicit formula given in the Encyclopedia: $$a_n=\frac{2n!}{\pi e}\Im\left(\int_{0}^{\pi} e^{e^{e^{ix}}} \sin(nx) dx \right)$$</p> http://mathoverflow.net/questions/97450/converting-a-recursive-definition-to-an-explicit-one/97454#97454 Answer by Steven Landsburg for Converting a recursive definition to an explicit one Steven Landsburg 2012-05-20T02:14:57Z 2012-05-20T02:14:57Z <p>You are computing the <a href="http://en.wikipedia.org/wiki/Bell_numbers" rel="nofollow">Bell numbers</a>.</p>