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How can we prove that

$$\sum_{r=1}^n \binom{2n}{2r-1}\frac{B_{2r}}{r}=\frac{2n-1}{2n+1}$$

where $B_{2r}$ are the Bernoulli numbers?

$$\begin{array}{c|c|c|} n & \frac{2n-1}{2n+1} & \sum_{r=1}^n \binom{2n}{2r-1} \frac{B_{2r}}{r} \\ \hline 1 &\frac{1}{3} & \frac{1}{3} \\ 2 &\frac{3}{5} & \frac{3}{5} \\ 3 &\frac{5}{7} &\frac{5}{7} \\ 4 & \frac{7}{9} & \frac{7}{9} \\ 5 & \frac{9}{11} &\frac{9}{11}\end{array}$$

The formula appears to be correct for a lot of values of $n$.

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  • $\begingroup$ Simpler answer obtained incidentally here: Because $B_0 = 1$, $B_1 = -\frac{1}{2}$, and $B_i = 0$ for all odd $i \geqslant 3$, $0 = \sum_{i = 0}^{2n} \binom{2n + 1}{i}B_i = \sum_{r = 0}^n \binom{2n + 1}{2r}B_{2r} + 1 - (2n + 1)/2$, whence $\sum_{r = 0}^n \binom{2n + 1}{2r}B_{2r} = n - \frac{1}{2}$, as in Marko Riedel's answer below. $\endgroup$ – Calum Gilhooley Mar 21 '15 at 1:25
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$$ \sum_{r=1}^n \binom{2n}{2r-1}\frac{B_{2r}}{r} $$ $$ = 2\sum_{r=1}^n \binom{2n}{2r-1}\frac{B_{2r}}{2r} = \frac{2}{2n+1}\sum_{r=1}^n \frac{(2n+1)(2n)!}{2r(2r-1)!(2n-2r+1)!}B_{2r} $$ $$ = \frac{2}{2n+1}\sum_{r=1}^n \frac{(2n+1)!}{(2r)!(2n-2r+1)!}B_{2r} = \frac{2}{2n+1}\left(\sum_{r=0}^n \left(\binom{2n+1}{2r}B_{2r} + \binom{2n+1}{2r-1}B_{2r-1}\right) - \binom{2n+1}{0}B_0-\binom{2n+1}{1}B_1\right) $$ $$ = \frac{2}{2n+1}\left(\sum_{r=0}^{2n} \binom{2n+1}{r}B_{r} - \binom{2n+1}{0}B_0-\binom{2n+1}{1}B_1\right) $$ Since, $$ B_m = -\sum _{k=0}^{m-1}\binom{m}{k}\frac{B_k}{m-k+1} \implies 0 = \sum _{k=0}^{m}\binom{m+1}{k}B_k $$ So, $$ \frac{2}{2n+1}\left(\sum_{r=0}^{2n} \binom{2n+1}{r}B_{r} - \binom{2n+1}{0}B_0-\binom{2n+1}{1}B_1\right)= \frac{2}{2n+1}\left(0-1-(2n+1)(\frac{-1}{2})\right) = \frac{1}{2n+1}\left(2n+1-2 \right) = \frac{2n-1}{2n+1} $$

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  • $\begingroup$ Thanks a lot! I see you used the fact that $B_n=0$ for odd $n$ greater than 1. $\endgroup$ – Shobhit Bhatnagar Jun 6 '14 at 10:33
  • $\begingroup$ @shobhit.iands yes. Forgot to mention that. $\endgroup$ – tpb261 Jun 6 '14 at 10:35
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This identity also has a proof using the technique of annihilated coefficient extractors (ACE).

First observe that it is equivalent to $$\sum_{r=1}^n \frac{2n+1}{2r} {2n\choose 2r-1} B_{2r} = n - \frac{1}{2}.$$

The left simplifies to $$\sum_{r=1}^n {2n+1\choose 2r} B_{2r}.$$

Introduce the following generating function $f(z)$ for this quantity, which is $$f(z) = \sum_{n\ge 1} \frac{z^{2n}}{(2n+1)!} \sum_{r=1}^n {2n+1\choose 2r} B_{2r}.$$

By the generating function of the Bernoulli numbers we have that $f(z)$ is $$\sum_{n\ge 1} \frac{z^{2n}}{(2n+1)!} \sum_{r=1}^n {2n+1\choose 2r} (2r)! [w^{2r}] \frac{w}{e^w-1}.$$

Switch summations to get $$\sum_{r\ge 1} \left( [w^{2r}] \frac{w}{e^w-1} \right) \sum_{n\ge r} \frac{z^{2n}}{(2n+1-2r)!}$$ which is $$\sum_{r\ge 1} \left( [w^{2r}] \frac{w}{e^w-1} \right) \sum_{n\ge 0} \frac{z^{2n+2r}}{(2n+1)!}.$$ This in turn simplifies to $$\sum_{r\ge 1} z^{2r} \left( [w^{2r}] \frac{w}{e^w-1} \right) \sum_{n\ge 0} \frac{z^{2n}}{(2n+1)!}.$$

The first term is the promised annihilated coefficient extractor and the second is $\sinh(z)/z$ so we get $$f(z) = \left(-1 + \frac{1}{2} z + \frac{z}{e^z-1}\right) \frac{\sinh(z)}{z}.$$

We extract coefficients from the three components. First, $$(2n+1)! [z^{2n}] \left(-\frac{\sinh(z)}{z}\right) = -(2n+1)! [z^{2n+1}] \sinh(z) = -1.$$ Second, $$(2n+1)! [z^{2n}] \left(\frac{1}{2} z \frac{\sinh(z)}{z}\right) = (2n+1)! [z^{2n}] \frac{1}{2} \sinh(z) = 0.$$ And third, $$(2n+1)! [z^{2n}] \frac{\sinh(z)}{e^z-1} = (2n+1)! [z^{2n}] \frac{1}{2}\frac{e^z-e^{-z}}{e^z-1}.$$ This last one needs some rewriting as in $$\frac{e^z-e^{-z}}{e^z-1} = 1 + \frac{-e^z + 1 + e^z - e^{-z}}{e^z-1} \\= 1 + \frac{1 - e^{-z}}{e^z-1} = 1 + e^{-z} \frac{e^z - 1}{e^z-1} = 1 + e^{-z}.$$ Therefore the third component is $$(2n+1)! [z^{2n}] \frac{1}{2} (1 + e^{-z}) \\ = (2n+1)! \times \frac{1}{2} \times \frac{(-1)^{2n}}{(2n)!} = (2n+1)\times \frac{1}{2} = n + \frac{1}{2}.$$

The sum of the three contributions is $$n + \frac{1}{2} + (0) + (-1) = n - \frac{1}{2}$$ precisely as was to be shown.

There is another annihilated coefficient extractor at this MSE link I and yet another one at this MSE link II.

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