Wikipedia:Reference desk/Archives/Mathematics/2013 May 4

= May 4 =

Explicit equations for these three sequences
I've looked at the pages for the up/down numbers, the Bernoulli numbers, and the Euler numbers linked to from the page of trigonometric functions, but I don't see the functions produce them. What are they? --Melab±1 &#9742; 03:49, 4 May 2013 (UTC)


 * Obviously there may be a function, but these numbers are described and the way the values are generated are by counting the instances. For n=0, there is only one permutation {}. For n=1 there is only 1 permutation {1}. For n=2, we must have c1c3 there are two possible permutations: 1,3,2 and 2,3,1 so An=2. For n=4, see the list on the Alternating permutation page. That is where 1,1,1,2,5,... come from. -- SGBailey (talk) 06:16, 4 May 2013 (UTC)
 *  Jack: It isn't bad spelling, just bad typing, there's a difference! sgb 
 * That's OK. I accept your spelling is not as bad as your typography would indicate.  --   Jack of Oz   [Talk]  01:15, 7 May 2013 (UTC) 
 * There are explicit formula for the last two Bernoulli number, Euler number and Up/down number can be easily derived from those. Indeed
 * $$A_n = i^{n+1}\sum _{k=1}^{n+1} \sum _{j=0}^k {k\choose{j}} \frac{(-1)^j(k-2j)^{n+1}}{2^ki^kk}$$
 * this might not be the best way to generate the numbers.--Salix (talk): 06:51, 4 May 2013 (UTC)