Trying to sum a series (related to catalan numbers perhaps) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:22:07Z http://mathoverflow.net/feeds/question/37253 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37253/trying-to-sum-a-series-related-to-catalan-numbers-perhaps Trying to sum a series (related to catalan numbers perhaps) David Willis 2010-08-31T11:52:42Z 2010-08-31T12:04:23Z <p>Whilst trying to solve a combinatorics problem I am faced with summing this series:</p> <p>1+ 2C_1 2/(3^2) + 4C_2 (2^2)/(3^4) + 6C_3 (2^3)/(3^6)+ ... + 2nC_n (2^n)/(3^(2n))+...</p> <p>Where 4C_2 is 4 choose 2.</p> <p>Any idea how to approach this problem?</p> http://mathoverflow.net/questions/37253/trying-to-sum-a-series-related-to-catalan-numbers-perhaps/37254#37254 Answer by Gjergji Zaimi for Trying to sum a series (related to catalan numbers perhaps) Gjergji Zaimi 2010-08-31T11:56:44Z 2010-08-31T11:56:44Z <p>The generating function of the <a href="http://en.wikipedia.org/wiki/Central_binomial_coefficient" rel="nofollow">central binomial coefficients</a> is $$\sum_{n=0}^{\infty}\binom{2n}{n}x^n=\frac{1}{\sqrt{1-4x}}$$ and so the value of your series is 3.</p> http://mathoverflow.net/questions/37253/trying-to-sum-a-series-related-to-catalan-numbers-perhaps/37255#37255 Answer by Martin Rubey for Trying to sum a series (related to catalan numbers perhaps) Martin Rubey 2010-08-31T12:04:23Z 2010-08-31T12:04:23Z <p>edit: the preceding answer suggests that my browser didn't display the dots, i.e. you really meant the series, not the sequence... Sorry, the below doesn't answer the question.</p> <hr> <p>Does the following look right? (might this be homework?)</p> <pre> (1) -> f n == reduce(+, [binomial(2*i, i)*2^i/3^(2*i) for i in 0..n]) Type: Void (2) -> guess([f n for n in 0..20], maxLevel==2) Compiling function f with type NonNegativeInteger -> Fraction( Integer) s - 1 21 8p + 12 ++-++ 20 4 | | --------- | | 9p + 18 n - 1 p = 0 20 --+ 20 (2) [ > ------------------- + 1] --+ 9 s = 0 21 Type: List(Expression(Integer)) (3) -> guessPRec [f n for n in 0..20] (3) [ [f(n): (9n + 18)f(n + 2) + (- 17n - 30)f(n + 1) + (8n + 12)f(n)= 0, 13 f(0)= 1, f(1)= --] 9 ] Type: List(Expression(Integer)) </pre> <p>In general, it's often a good idea to generalise, i.e., introduce more parameters:</p> <pre> (4) -> f n == reduce(+, [binomial(2*i, i)*x^i/y^(2*i) for i in 0..n]) Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f Type: Void (5) -> guess([f n for n in 0..20], maxLevel==2) Compiling function f with type NonNegativeInteger -> Fraction( Polynomial(Integer)) s - 1 21 (4p + 6)x ++-++ 20 2x | | ----------- | | 2 n - 1 p = 0 (p + 2)y --+ 20 20 (5) [ > ---------------------- + 1] --+ 2 s = 0 y 21 Type: List(Expression(Integer)) (6) -> guessPRec [f n for n in 0..20] (6) [ [ f(n): 2 2 (n + 2)y f(n + 2) + ((- n - 2)y + (- 4n - 6)x)f(n + 1) + (4n + 6)x f(n) = 0 , 2 y + 2x f(0)= 1, f(1)= -------] 2 y ] Type: List(Expression(Integer)) </pre>